Bayesian evaluation for the likelihood of Christ's resurrection (edit 1)

I'm working on editing Bayesian evaluation for the likelihood of Christ's resurrection.

Last week and this week, I wrote out the table of contents, and put in the skeleton around which the text can be inserted. I also copied some text over so that I can get an idea of what things will look like.

You may next want to read:
Sherlock Bayes, logical detective: a murder mystery game
Key principles in interpreting the Bible
Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection

This post will grow and change to eventually contain the entire series on the resurrection.

This is a work in progress. It will take some time to complete, and it will be messy in the meantime. The contents here will change as I add in the previous posts, make edits, and add some additional material which I didn't get a chance to mention.

Outline / Contents:

PART I: The evidence for the resurrection completely overwhelms the prior.

Chapter 1: The basic argument:
- The prior odds against a resurrection (part 2)
- The value of a human testimony. (part 1, 3)
- The evidence for a resurrection - is it enough? (part 4)
- Conclusion: there is in fact far more than enough evidence to overcome the prior.

Chapter 2: Double checking the strength of a human testimony: (part 12)
- Anatomy of a human testimony.
-- License plate effect (parts 25, 26, new material)
-- Dependency factors (part 27, 28)
- More examples establishing the Bayes' factor of a human testimony.
-- Car accident (part 13)
-- human death (part 13)
-- LinkedIn claim (part 14)
-- 9/11 false claims (part 15)
-- One in a million events happen every month (new material)
-- Other reports of a resurrection in history (part 30)
-- Video of a lottery winner (part 26)
- revisiting our initial calculation.
- Conclusion: The strength of a human testimony is firmly established and understood. (part 16)

Chapter 3: Double checking our conclusion against other reports of resurrections in history.
- Rationale: Are there other historical cases of reported resurrections? How good are they? (part 17)
- The conditions: the requirements for "matching" a testimony. (part 18)
- The other historical records:
-- Apollonius of Tyana (part 19)
-- Zalmoxis (part 20)
-- Mithra (part 21)
-- Horus (part 21)
-- Osiris (part 21)
-- Dionysus (part 21)
-- Krishna (part 22)
-- Bodhidharma (part 23)
-- Puhua (part 24)
- Conclusion: Our previous calculation is fully validated. (part 29, 30)

PART II: Answering simplistic objections

Chapter 4: Simplistic objections:
- Is the prior too large, especially for a supernatural event? (part 5)
- Can human testimonies be trusted?
-- Could the disciples have been genuinely mistaken? (part 6)
-- Or actively deceptive? (part 7)
-- Or actually crazy? (part 8)
-- Or some combination of the above, or something else entirely? (part 10)
- Conclusion: there are no reasons to consider any of these. And even so...

Chapter 5: The strength of a Bayesian argument: why none of these objections work.
- Everything is already taken into account. (part 9)
- Only evidence moves the odds. Speculations do nothing. (part 10)
- The lack of evidence against the resurrection. (part 11)
-- crackpot theories, as suggested by "disciples stole the body" rumor, to be addressed later.
- Conclusion: The Bayesian argument already takes it all into account.

PART III: Even crackpot theories are not enough.

Chapter 6: Addressing all possible theories - including crackpot theories
- recall: crackpot theories, like "disciples stole the body" were to be addressed later. (part 11)
- not much choice left now (part 30)
- Examining crackpot theories, in general. (part 31)
- Conclusion: we now turn to examining even such theories.

Chapter 7: The "skeptic's distribution" approach.
- Use the historical data to construct the skeptic's distribution. (part 34)
-- properties of the skeptic's distribution: power law. (part 35)
-- details of the distribution: generalized Pareto distribution and its parameters (part 36)
-- Having more "outliers" similar to the maximum makes a true outlier far less likely (part 37)
- The calculation: spell out the details of the program (part 38)
-- Simulation and code: The number of "outliers" decides the case. (part 39)
-- The list of 50 outliers. This puts the Bayes' factor beyond the prior. (part 40)
- Conclusion. There resurrection hypothesis still holds.

Chapter 8: The generosity in the "skeptic's distribution" approach.
- The more likely adjustments. We were far too generous to the skeptic. (part 41, 42)
- The same kind of calculation for youtube videos and their view counts (new material)
- The simulation and code (part 43), with the new adjustments.
- Anti-crackpot theory defenses built in to Christianity. (part 32, 33)
- Conclusion: the resurrection is still certain, even after taking crackpot theories into account. (part 44)

Chapter 9: One more double check: validation through miracles in other religions. (part 47)
- Vespasian (part 48)
-- "something happened" vs. "a miracle happened". (part 48)
- Ichadon (part 50)
- Splitting the Moon (part 50)
- Accounts in Josephus
-- Honi the Circle-drawer (part 49)
-- Eleazar the exorcist (part 49)
- even with just 10x the level of evidence, with no increase in outliers, it's enough (new material)
- Conclusion: this validates our approach. nothing approaches the level of evidence for the resurrection.

PART VI: Final Challenge and conclusion

Chapter 10: The final challenge: replicate the results. (Part 45, 46)

Chapter 11: Conclusion and epilogue (Part 51, 52)


The basic argument

Chapter 1:
Priors and evidence

The prior odds against a resurrection

What is the probability that Jesus rose from the dead?

Here I'm going to construct a foolish partner to advance certain arguments. This is just a rhetorical device. I have to be careful to not commit a straw man here, nor do I wish to insult anyone. I don't intend to imply that anyone actually thinks like my partner. But while he's too foolish to actually stand in for any real person, he can therefore be useful, by standing in as the lower bound on what a reasonable person may think. Please, just understand him as the artificial rhetorical construction that he is.

Now, my foolish partner may say, "the probability that Jesus rose from the dead is zero. What's there to talk about?" But by doing so, he has committed the cardinal sin in Bayesian reasoning. Any real, non-theoretical probability CANNOT be absolutely zero or one. Think about what a zero probability value means: this represents a state of mind where absolutely NOTHING - no amount of even theoretically possible evidence - can alter their beliefs. There is no possible reasoning with such a person.

I am very certain that the sun will rise tomorrow. I may be 99.9999...% certain, but I cannot be 100% certain. That tiny difference between 99.9999...% and 100% represents possibilities like a super-advanced alien race stopping the rotation of the earth, or me being momentarily confused about what is meant by "the sun". And I am not 100% certain, because I can, at least in theory, be shown evidence that such an alien race exists, or that I had momentarily confused "the sun" with "the north star".

My partner may then say, "well, the probability may not be actually zero, but it's very close to it. Like, 0.000.....001%. Nobody has ever come back from the dead before." But actually, isn't that the very thing we're talking about? Whether Jesus had come back from the dead? Furthermore, it's presumptuous to think Jesus was just like everyone else, that he wasn't special in any way. Even if nobody else came back from the dead, we would need to do some additional thinking in the case of Jesus.

My partner would reply, "see, that's just special pleading. I don't see why Jesus should be special. Empirically, people do not come back from the dead. Therefore it's also highly unlikely that Jesus came back."

At this point, I'm going to simply give away the point about whether Jesus was special or not. I obviously believe that he was - but quite frankly, the argument for the resurrection is so strong that I can just handicap myself in several different ways like this without materially affecting the conclusion. I'll be doing this multiple times throughout this post.

Now, let's talk about how many people came back from the dead, "empirically". How many different people have you seen die and stay dead? Remember, we're talking about "empirical" evidence here, meaning that we only count people that you, yourself, have seen die in person. For many people, that number is probably zero. It might be one or two - maybe you've seen a grandparent pass away. Maybe more, if you're a health care worker or something like that.

My partner may say, "Even if I didn't see someone die in person, if there was a real resurrection, it would be all over the news. And there hasn't been any such reports, because people do not rise from the dead."

Well, at this point, my partner is begging the question on whether there has in fact been such reports, and is becoming slippery about what "empirically" means. But again, I will simply handicap myself and give away these points. "Empiricism" in the sense of "I only believe what I can see" is fundamentally flawed, anyway (It's self-defeating). So let's say news reports are enough, that a direct observation is not necessary. So, how many people have been covered in the news that you've seen? Thousands? Millions? If the argument is that Jesus was no different than these thousands or millions of other people, then I freely acknowledge that this does in fact establish an upper bound on the probability of the resurrection. However, this does NOT prove that the probability is zero, no more than a dozen coin flip of heads proves that the coin will always land heads. Instead, it merely says that the probability for the resurrection is likely to be below a certain level.

For example, say that you've examined a thousand swans, and they all turned out to be white. You want to use this fact to investigate the report of a black swan. Now, your thousand white swans don't prove that the probability of the reported swan being black is zero. Instead, combined with that report, it does say that the probability is likely to be below 1/1000. If you've examined a million swans, and all of them have been white, then your probability of observing a black swan would correspondingly drop to around 1/1 000 000 as the upper limit.

Now, the modern media is pretty comprehensive, so my partner may say, "The world news covers at least millions of other people. And none of them have come back from the dead. So the chance that Jesus came back from the dead is, at best, one in a million. That's basically zero. How could you believe in something that has only one in a million chance of being true? That's irrational."

Well, one in a million is a pretty small probability. But actually, I think we can just go ahead and say that out of the entire world population of 7 billion people, none of them are going to be raised from the dead. So, the probability for the resurrection has now dropped to 1 in 7 billion. I'm just giving away everything here. I've almost dropped the condition about an "empirical" probability. I'm making a blanket statement that absolutely nobody in the world, independent of anything that may be know about them, will rise from the dead. So, if we apply this general "observation" to the likelihood of Jesus's resurrection, that probability must be below 1 in 7 billion.

My partner may respond, "Um... So now you're making my argument for me. So yeah. The probability of the resurrection is less than 1 in 7 billion. Obviously you can't believe in something that unlikely to be true. This is why any naturalistic explanations must always be preferred to a supernatural one in these discussions of miracles, because the supernatural is always so unlikely."

Oh, but I'm not done yet. I'm going to give away even more of the argument. Why not just drop all pretense of an "empirical" probability? Why not say that everyone who has EVER lived - about 100 billion people in total - have all died, without a single one of them being raised from the dead? Forget saying anything about "empirical observations". Forget any semblance of reasoning from our direct experiences. I will simply grant that every single one of these 100 billion people have died and stayed dead. And against the weight of those 100 billion people, we'll estimate the probability of Jesus's resurrection. According to our previous line of thinking, this puts that probability at 1 out of 100 billion.

My foolish partner may say, upon the strength of this evidence that I have made up for him, "One in a hundred billion! Do you know how unlikely that is? That's 1 out of 100 000 000 000. That's a probability value of 0.000 000 000 01. That's basically zero. Just concede the argument - it's virtually impossible that Jesus rose from the dead. Absolutely any naturalistic explanation is going to be more likely than that." Oh, but I'm not done yet.

I'm going to give away another multiplier in the probability. I'm going to make it even smaller - not by an additional factor of ten, or even by a factor of a thousand. No, I'm going to give away far more. I'm going to SQUARE that tiny previous probability of 1 in 100 billion, and use that as the probability of Jesus's resurrection. One in a hundred billion, squared, is this:

probability = 1/10 000 000 000 000 000 000 000, or 0.000 000 000 000 000 000 000 1

There is no reason to do this. Squaring the probability makes no rational sense. I did it just to make the probability smaller, to handicap my argument further. I started with the "Nobody rises from the dead. It's never happened before. So Jesus also didn't rise from the dead" argument. I then stretched it to its strongest form, then started making stuff up to make it even stronger. I then ran out of stuff to make up, but I still wanted it to be even stronger - so I simply squared the already tiny probability value, with no possible rationale, to arrive at the absurdly minuscule probability value above. So now, as it stands, the probability of Jesus actually having risen from the dead is 1 out of 10 000 000 000 000 000 000 000 - essentially zero. That's game over, right? How could I, or anyone, believe in something so unlikely to be true? How could any hypothesis with a probability of 0.000 000 000 000 000 000 000 1 ever be taken seriously?

"Um... so yeah. What are you doing?", my partner may ask.

You'll see. Next week, you'll behold and understand the power of evidence.

The value of a human testimony

Let's say that you're meeting someone new. You talk for a while, and the conversation turns to birthdays. You reveal that you were born in January, and your new friend says, "Oh, really? I was born in January too!" He seems earnest - he's not obviously joking, sarcastic, or ingratiating. From the little you know of him, he's not any more likely to be delusional or deceptive than anyone else.

Now, based only on his earnest word, would you be willing to believe that your new friend really was born in January? Note that I'm not looking for 100% certainty here. A willingness to entertain the idea, to give it at least a 50-50 shot of being true, is all that's required.

Also note that I'm not asking whether this event is likely to happen. Obviously, the probability that you and a random other person shares the same birth month is about 1/12, so it may be said to be "unlikely". Rather, I'm asking whether you would believe this person, given that this unlikely event has already occurred.

So, how would you respond? Would you say, "I find your claim to be highly dubious. There's only 1/12 chance that you were born in the same month as me"? Or would you simply reply, "Oh, hey, that's neat!"

I'm going to assume that you're willing to believe your new friend. I think you'll agree that it takes a special kind of jerk to say "I don't believe you. You must be lying or mistaken. It's just too unlikely for us to share the same birth month". In that case, what if it turns out that you share the exact same birthday? You mention that you were born January 23rd, and he claims the same. Would you still believe him?

Let's continue the same line of thought: what if you then tell him your mother's birthday, and what do you know - it turns that the date is also his mother's birthday! Your fathers, too, turn out to share a birthday. "Wow", he says, "so the three members of our family all share the same birthdays - amazing!" Would you be willing to believe him on this?

If so, at what point in comparing family birthdays would it become too unlikely for you to believe? That is, if you continued on to compare your grandparents and uncles and cousins, and they all continued to have the same birthdays, at what point would you say "I cannot believe this - this is too unlikely to be true", in spite of your friend's earnest insistence?

Decide on an answer, and remember it. Write it down somewhere. We'll come back to this answer in the coming weeks. Make a firm statement like, "I would be willing to believe up to 3 shared birthdays - myself, mother, and father - but if he claims 4 or more shared birthdays I would begin to be skeptical".

Let's try another example. Let's say that you run into an acquaintance whom you haven't seen in a while. You exchange greetings and ask how he's been recently, and he excitedly tells you - "Guess what! I've actually won the jackpot in the lottery last month! I'm rich!" As before, he seems earnest - he's not obviously joking, sarcastic, delusional, or deceptive. Would you believe him, based only on his earnest word? Again, only a willingness to entertain the idea, just granting a 50 - 50 chance of it being true, is all we're looking for here. Would you give him at least even odds that he's telling the truth?

And if you would, how about if he claims to have won two consecutive jackpots? How about three? At which point would you say "That's just too much for me to believe"?

Next, let's switch over to other gambling games. Say that a friend claims to have had a very lucky night at the card tables. He says that he got a royal flush in a 5-card stud poker game. Would you believe him? What if he claims to have gotten two royal flushes last night? What if he claims three? At what point would you say, "I don't believe you. You seem earnest and all, but the chances of that happening are just too small"?

How about if he were playing Texas Hold'em, and claims to have had multiple pocket aces? Say that he claims to have had two, three, four, or five pocket aces last night. At what number does it become too unlikely to be true, despite your friend's earnest claim?

We can ask similar kinds of questions in many different ways. What if someone claims to be born as a part of twins, triplets, quadruplets, or quintuplets? What if someone claims to have recently been struck by lightning? Or that they were a victim of two or three such strikes?

Remember, in all these cases, that we're not looking for certainty. Just a willingness to grant even odds - a 50-50 likelihood for the statement is true - is enough to say that you'd believe your friend. Also, we're not asking whether these scenarios are likely; rather we're asking if you'd continue to believe this earnest person, despite the fact that he's claiming that an unlikely event happened.

Answer these questions. Give a specific number in each case: we want answers like "four royal flushes" and "two lightning strikes". Write them down somewhere - we'll come back to them later.

Next week, we'll turn to the question of Jesus's resurrection.

The previously given probability value for the resurrection - 0.000 000 000 000 000 000 000 1 (which can also be written as 10^-22, or 1e-22) - is a prior probability. That is, it's the probability based on the background information, taking into consideration the fact that Jesus was human, and that humans don't rise from the dead.

However, it is just the starting point. It does not take into account any evidence we have specifically about Jesus's resurrection. Remember Bayes' rule: the final, posterior odds is the prior odds times the likelihood ratio. The number we have now is just the prior odds. We now need a numerical value for the likelihood ratio of the evidence, and then we can get our posterior odds.

But what kind of evidence is there for Christ's resurrection? And how could it possibly overcome a prior odds of 1 to 10 000 000 000 000 000 000 000 against it? Well, as for the evidence, we have the writings of the New Testament, where Jesus's resurrection and his follower's testimonies are documented. Okay, but is this "evidence" any good? How can we put numerical likelihood ratios to these things?

What we need is the numerical strength of a human testimony. As it turns out, we can actually get a not-unreasonable, order of magnitude estimate of this value. Remember your answers to the probability questions at the beginning? I hope you have them written down or otherwise recorded, because we will use them to calculate the likelihood ratio value that you would personally assign to a typical human testimony.

Let's use my personal answers, given below, as an example for how to do these calculations. These are my gut answers to the questions, before doing an actual probability calculations. Remember, these are the events that I'm willing to give even odds (50/50 chance) on, based solely on an earnest, personal testimony. It does not mean that I'm willing to believe 100%, and it does not mean that I'd stop looking for more evidence. It only points to how much I'm willing to adjust my beliefs based on someone saying "yes, I know it's unlikely, but it really happened".

For the shared birthday question, I would easily believe that my friend shared a birthday with me. I would also not have any real problem believing that our mothers also shared birthdays. At three people - myself, mother, and father - I would start becoming skeptical, but would probably give my friend the benefit of doubt. Starting with four shared birthdays in the family, I would start leaning more heavily towards skepticism.

On winning the lottery, I would not really doubt that my friend won the lottery. I would start doubting if he says that he won two consecutive lotteries.

On getting a royal flush, I think I could almost believe that my friend got two such hands in a very lucky night at the table. I feel like three would be entering the realm of the fantastical, and I would doubt my friend at around this number.

On pocket aces, I would be willing to believe that my friend had up to four or five pocket aces in a lucky night of Hold'em.

On the multiple births, I would not have any real problems believing that someone was a part of quadruplets. A claim to be in a quintuplet would start to cause a little bit of doubt to me, and a claim of sextuplets would need additional evidence.

On being struck by lightning, I actually had someone around me claim that this had recently happened to her. I had no problem believing it. Even if she had claimed two such accidents I don't think I would have really doubted her. If she had claimed three, I would start to be skeptical.

Now, calculating the numerical probability values for all these things is pretty straightforward:

The probability of sharing a single birthday is 1/365, or 1/3.65e2. The probability of sharing the three birthdays for your family is then simply this number cubed - 1 in 4.86e7.

The probability of winning the lottery varies by exactly which lottery you're talking about, but the odds for the jackpot are generally somewhere around 1 in 1e8.

The probability of getting a single royal flush is 1 in 6.5e5. The probability of getting two in two hands is therefore this number squared, 1 in 4.2e11. We can then take it down by a couple orders of magnitude, to account for the fact that there's dozens of hands played in a poker night. That gives us something like 1 in 1e9 for the odds.

The probability for getting pocket aces is 1 in 221. Getting five would then be 1 in 5.3e11. Taking it down again by several orders of magnitude to account for multiple hands, that brings us to something like 1 in 3e7.

The probability of quadruplets is about 1 in 1e6, and for quintuplets it's about 1 in 5.5e7. We'll split the difference here and call it 1e7.

The probability of getting struck by lightning in a given year is about 1 in 1e6. If we count "recently" as the last 5 years, that would bring it down to 1 in 2e5. Getting struck twice would then be 1 in 4e10, then maybe take off an order of magnitude for possible dependency factors to give us 1 in 4e9.

So, looking at the final numbers above - 1/4.9e7, 1/1e8, 1/1e9, 1/3e7, 1/1e7, 1/4e9 - we seem to be getting a reasonably consistent estimate for how I value the strength of an earnest, personal testimony. There are a lot of small details we can go over again (how many hands of poker did you play last night? Is your friend someone likely to play the lottery, or to be outdoors during a thunderstorm?), but these will largely be random, small, unknowable effects that will get washed out in this order-of-magnitude calculation.

So, we'll take the geometric mean of the above values(1/7e9), and then conservatively knock off a couple orders of magnitude, to get 1/1e8 as their "average" probability. In other words, even if an event had only a 1/1e8 prior chance of happening, I would be willing to give even odds on that event having occurred based on someone's earnest, personal testimony.

At such small probability values, "probability" is nearly synonymous with "odds". Therefore, I can re-state the above as saying that an earnest, personal testimony will shift the odds from 1/1e8 to 1/1. Or, to put it yet another way: the typical Baye's factor for an earnest, personal testimony is around 1e8. That is my numerical value for the strength of a human testimony.

It is important to note that this number is not something that I just made up. The math that gives this value is described above in its entirety. What answer did you get when you plugged in the numbers? That is the number that you, personally, must be willing to assign to the strength of a personal testimony, in order to be consistent. I believe that most reasonable people will be within a couple of orders of magnitude of my answer.

Furthermore, this number is something that you can check for yourself, based on a thought experiment that you can perform on yourself. Imagine a future where you yourself are telling someone, "I just hit the jackpot in the lottery". You are being earnest and sincere. Now, what is the probability that you're telling the truth in your own hypothetical future?

Given that the odds of winning the lottery is about 1/1e8, if you agree with my assessment that personal testimony should be valued at a Bayes' factor of around 1e8, then you are about equally likely to be telling the truth or lying in this scenario. However, if you disagree with that assessment - for example, if you think that personal testimony should only be valued at 1e6 - then you're saying that the posterior odds of you having won the lottery is still only 1/100, and so you're 99% likely to be lying in that scenario. Which is it?

In fact, this thought experiment suggests a way to empirically verify this value. Simply investigate a random sample of the people who claimed to have won the lottery. Remember, we're only counting earnest, personal claims to the jackpot. What fraction of them are telling the truth? How many of them are actual lottery winners? If you say "maybe around half?", then you're agreeing with my Bayes' factor of 1e8. If you want the Bayes' factor to be 1e6 instead, then you need 99% of these people to be liars.

Do you still doubt that you can assign a numerical value to the strength of a personal, human testimony? Or maybe worry that the correct value is far from 1e8? Well, fortunately for us, this "lottery liars" experiment has actually been naturally conducted, and we can compare its result with my numbers.

On January 13, 2016, the Powerball lottery produced the largest jackpot in history to date: 1.6 billion dollars. This jackpot ended up being split three ways. But - were there people who lied about having won this jackpot? As a matter of fact, there were. Several people on social media claimed to be a winner, presumably in an attempt at some quick, cheap fame. How many such people were there?

I couldn't get an exact number for the number of Powerball jackpot liars, but we can still get a sense, an order-of-magnitude estimate. Snopes, for example, mentions two people by name, and "several" or "numerous" others. Another report claims "a number" of similar hoaxes. So - it sounds like maybe ten people lied about winning the jackpot? It's certainly not in the hundreds or thousands.

How does that compare with the estimates from my probability calculation? Well, the odds of hitting the jackpot in Powerball are about 1/3e8. However, people may buy multiple tickets - which many people certainly did on such a well-publicized jackpot. In the end, there were 3 actual winners, out of the total American population of 3e8 people. So the prior odds for a specific person in the United States being a winner was 3/3e8, or 1/1e8.

Now, if the Bayes' factor for an earnest personal testimony is 1e8, then the posterior odds is just the product of 1/1e8 and 1e8, which is 1. That translates into 1 actual winner for every liar. So, given that there were 3 actual winners to the jackpot, we should expect around 3 liars - and that is roughly what we actually appear to have, within an order of magnitude.

You can again nitpick at this example (the great publicity of this jackpot, the people who made an earnest claim offline, the relative certainty of a short-lived notoriety for lying, etc.) But as an order-of magnitude estimate, the results of this natural experiment are about as good as I can possibly hope for. So, the proper Bayes' factor for an earnest, personal testimony is typically about 1e8, and this has now been validated through multiple lines of thought. It is certainly not several orders of magnitude less than that.

Next week, we will therefore continue the rest of the calculation using 1e8 as the Bayes' factor.

The evidence for a resurrection - is it enough?

Now that we have all the necessary numerical values, we can finally calculate the probability that Jesus rose from the dead.

To begin, I gave the prior odds for Jesus's resurrection as 1e-22. This number was obtained from the argument that "empirically, people do not rise from the dead. Therefore, Jesus also couldn't have risen from the dead." I took that argument, then made it as strong as possible, then gave away everything that it asked for, then gave away some more things that it didn't ask for, then finally I strengthened it beyond all bounds of reason, by squaring an already tiny prior probability with no possible justification. In other words, this 1e-22 is a far smaller probability than anything that any skeptic can rationally ask for.

Next, we calculated a typical value for Bayes' factor, for a seemingly earnest, sincere, personal testimony. It worked out to be about 1e8. It's certainly not much less than that in the general case.

Now, we simply apply Bayes' rule: posterior odds are prior odds times Bayes' factors (the likelihood ratio). So, we'll just look through the New Testament, and see if we can find people who made an earnest, personal claim that Jesus rose from the dead. Let's start in 1 Corinthians 15, because that's a famous passage on the resurrection, recognized even by skeptical scholars as originating within a few years of Jesus's death. The passages reads:
For I delivered to you as of first importance what I also received: that Christ died for our sins in accordance with the Scriptures, that he was buried, that he was raised on the third day in accordance with the Scriptures, and that he appeared to Cephas, then to the twelve. Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep. Then he appeared to James, then to all the apostles. Last of all, as to one untimely born, he appeared also to me.
So, who in this passage can be said to have made an earnest, personal claim of Jesus's resurrection? Well, there's Cephas, also known as the apostle Peter. He's a major character in the New Testament, and every one of the numerous accounts of him says that he did, in fact, testify that Jesus rose from the dead. Certainly, that's one witness. The odds of Christ's resurrection after taking Peter's testimony into account is now 1e-22 * 1e8 = 1e-14.

Anyone else we can find here? Well, there's James, the brother of the Lord - the next named witness. He's another major character in the New Testament, another major player in early Christianity. We have no doubt that he professed that Jesus rose from the dead. So we have another witness. The odds of Christ's resurrection after taking James's testimony into account is now 1e-14 * 1e8 = 1e-6.

And then there's Paul, the author of the very passage we're reading, and one of the most prolific writers of the New Testament. He himself says in this very passage that the risen Christ appeared to him. The odds of Christ's resurrection after taking Paul's testimony into account is now 1e-6 * 1e8 = 1e2, or 100 to 1 FOR the resurrection.

Huh, would you look at that. After taking just three witnesses into account, the odds are now in FAVOR of the resurrection. And this is literally just a fraction of the way into the first passage we chose in the New Testament! Even within this passage, we still haven't taken into account the other members of the twelve disciples, or the other apostles, or the five hundred that are mentioned. And then, we still have the rest of the New Testament to still go through!

What happened? The prior odds was 1e-22 - that's 1 in 10 000 000 000 000 000 000 000! Wasn't that suppose to be an impossibly small odds? Wasn't it suppose to be insurmountable? Wasn't it something that enabled atheists to simply say, "therefore any naturalistic explanation is bound to be more likely"? Wasn't it a bulwark for skepticism, based on some kind of empiricism? How could it have just... evaporated like that?

That is the power of evidence. Evidence can cause swings in probability that seem ridiculously large to people who are not actually familiar with the mathematics. Did you think that a billion is a large number, or that a probability of one in a billion is too small to ever care about? It is not. In some kinds of math, even numbers like a googol (1e100) can disappear to nothing in just a few lines of calculation. And probability is one example of that kind of math.

Just the other day at my work (Bayes's theorem and probability calculations are part of my day job), a Bayes' factor of 1e-10 came up. It merited no comment beyond "that's pretty small". Another time, 1e-40 appeared as a Bayes' factor, again with little commentary on its magnitude. Numbers like that are not atypical in probability calculations. Do you realize that, if I specify the order of cards in a shuffled playing deck, that I'm doing so against an odds of 1 to 8e67? That if I hand you a record of a chess game (which can fit in a single post-it note), I'm specifying one out of at least 1e120 possibilities? So, a billion - which is only 1e9 - is not a large number. And the prior odds against the resurrection - which is only 1e-22 - gets completely blown away when it's set against the evidence.

Here, it's important to again note how much I'm handicapping the argument for the resurrection. I already mentioned how the prior probability of 1e-22 was far smaller than anything that a skeptic can reasonably ask for. As it turns out, the Bayes' factor of 1e8 for a personal testimony is also far smaller than it could have been. It's the right value for the general case, but in specific situations it may be far, far larger. Note the above example of recording a chess game: if you choose to believe that my record of the game is accurate, you're giving me a Bayes' factor of around 1e120 for my testimony. So that 1e8 really represents only the lower bound.

So, as it stands for the moment, the odds are 100:1 in FAVOR of the resurrection, using only Peter, James, and Paul's personal testimonies. The seemingly strong "nobody rises from the dead, so Jesus couldn't either" argument has been fully overcome, using absurdly conservative probability values, with only a tiny fraction of the evidence we have in the New Testament. At this point, the resurrection is already quite probable - but I suppose we might as well finish off the passage we've started on, to see how the odds grow from here.

As before, I'm going to be giving away multiple orders of magnitude in the following calculations, because the case for the resurrection is just that strong. I'm actually going to be somewhat sloppy about exactly how much I'm giving away, because it just does not really matter in the end.

So, let's see who else comes up in 1 Corinthians 15. It says that Jesus appeared to "the twelve", and also to "all the apostles". Now, it's clear that "the apostles" refer to a larger group of people than "the twelve", but let's just ignore that - we'll just say that these both refer to the twelve disciples. Furthermore, we'll go ahead and cut down this group even more, to include only those disciples who are mentioned more often in the New Testament. Say that leaves us with 6 disciples. With some dependency factors and all, let's give each of these disciples a Bayes' factor of 300 for their testimony. That value is far smaller than the 1e8 that we used earlier, and represents an extremely low opinion of their trustworthiness: you wouldn't believe such a person even if they told you their own birthday.

Well, even with these absurdly low estimates, the overall Bayes' factor is still 300^6, or about 1e15. The odds of Christ's resurrection, after taking into account the disciples' testimonies, is now 1e2 * 1e15 = 1e17.

1 Corinthians 15 also mentions Jesus appearing to "more than five hundred brothers at one time". It's clear that Paul had a specific set of people in mind, as they are part of this early central creed, and Paul mentions that some of these people have died. The number 500, too, is not something anyone just made up - it seems as if the passage is extra careful to mention that some have died, because this may have reduced the actual number of living witnesses to below 500. But let's just ignore all that. Let's pretend that Paul (and the early Christians) exaggerated this number by a factor of ten, so that there were only 50 people claiming to have seen the resurrected Christ. Let's furthermore give them a Bayes' factor of 2 for their testimonies - meaning, you trust them so little that you would hardly believe them when they tell you their own gender. Again, even with these absurdly low values, their overall Bayes' factor is 2^50, or 1e15. The odds of Christ's resurrection, after taking these people's testimonies into account, is now 1e17 * 1e15 = 1e32.

So, that brings us to the end of the 1 Corinthians 15 passage. We can go through the remainder of the New Testament, but that's a lot of work to improve an odds that's already at 1e32 - so this is a good place to stop for now. What have we achieved? Consider:

- We have only used the strength of personal testimonies. That is, we've only used the fact some people have said that they have personally witnessed to the resurrected Christ. We have not yet taken into account any other kinds of evidence, such as the fulfillment of Old Testament prophecies, or physical facts like the empty tomb, or historical facts like Christianity's explosive early growth, or anything else.

- We have used extremely conservative numbers in each step of our calculations, to the point of irrationality in some places.

- We have only focused on a single passage from the entire New Testament.

And even under these extreme conditions, the odds have easily overcome the 1e-22 prior odds against people in general rising from the dead, and are already at 1e32 to 1 for Christ's resurrection. If I were to carry out a more full and reasonable calculation, using all the different lines of evidence that a modern Christian has at his or her disposal, I do not doubt that the odds would turn out to be far in excess of 1e100. Jesus almost certainly rose from the dead.

Next week, we'll examine some possible objections to this calculation.

Conclusion: there is in fact far more than enough evidence to overcome the prior.


Chapter 2:
Double checking the strength of a human testimony

The rationale for double checking

My claim, at its heart, is very simple: the evidence of the many people claiming to have seen the risen Christ is abundantly sufficient to overcome any prior skepticism about a dead man coming back to life. My argument consists of backing up that statement with Bayesian reasoning and empirically derived probability values.

The emphasis on empirical probability values is important. Humans are notoriously bad at estimating probabilities, especially when the values reach extreme levels, like 1e-22. Some people, especially when discussing a controversial topic like the resurrection, will just pull numbers out of thin air to support their preconceptions. They'll make statements like "I'll grant a 23.599% chance that the disciples went to the wrong tomb". This can sometimes result in some pretty hilarious statements, like someone assigning a 1% chance for a generic conspiracy theory - as if they couldn't imagine anything less likely than a 1% probability.

This is why having an empirical bases for the probability values is crucial. Otherwise, you're likely to simply make up such worthless numbers, influenced only by your preconceived notions.

In my argument, none of the numbers I used are something I just made up. I gave each of them ample empirical backing. The two important numbers are the prior probability for the resurrection, and the Bayes' factor for a human testimony. I set the prior probability at 1e-22: this is, as I said, far more conservative than any requirement of empiricism. One may be able to empirically argue that nobody alive today has ever seen a man come back from the dead - this would set the prior odds at around 1e-9 or 1e-10. But I've gone much, much further. I've chosen the value of 1e-22 by taking the total number of all the humans that have ever lived, assumed that none of them have ever come back from the dead, then squaring the already tiny probability, just to handicap my argument further. There is no way to argue that it should be empirically set lower.

As for the Bayes' factor for a typical human testimony, I've set at 1e8. I've given numerous lines of thought that demonstrates that this is about the correct value. These including several examples from everyday life where you choose to trust someone, and the results of a natural experiment with the recent 1.6 billion dollar lottery. All these empirically derived lines of thinking converge around 1e8 as the correct value for the Bayes' factor of a typical human testimony.

But, this number is perhaps more difficult to accept than the prior probability. There is a large variance inherent in human testimony, and Bayes' factors are less familiar and less intuitive as a concept than a prior probability. For these reasons, I think it's worth demonstrating with a few more real-life examples that the Bayes' factors for a human testimony is really around 1e8.

We'll look at these examples in the coming weeks.

Anatomy of a human testimony: The license plate effect

(parts 25, 26, new material)

Dependency factors:

(part 27, 28)

More examples establishing the Bayes' factor of a human testimony.

Here are some more examples from which you can estimate the Bayes' factors for an earnest, personal human testimony.

Car accidents:

Imagine that you've promised to meet me on a particular date, but I don't show up to the appointment. You're understandably peeved, but then you get a phone call from me saying, "I just got into a car accident. I'm okay. But I'm really sorry that I couldn't make it to our meeting today. Can we still meet?"

Now, would you believe my story? Did I really get into a car accident on the day of our appointment? What would you assign as the probability that I'm telling the truth?

The average driver gets into a car accident roughly once in 18 years. That's about once every 6500 days. So the prior probability for getting into a car accident on a particular day is 1/6500. If you choose to believe me - say, you think there's less more than a 90% chance that I really was in an accident - then you've changed the odds for my car accident from 1/6500 to 10/1, and you've therefore granted my phone call a Bayes' factor of 65000 - or nearly 1e5.

Remember our calculations from earlier: even with a Bayes's factor of just 1e4, there's already a 99.999% chance for the resurrection to be real. In other words, if you would believe that I got into a car accident, you ought also to believe in the resurrection. Otherwise you're being inconsistent. If you wish to disbelieve the resurrection, you must also be the kind of person who says, "I don't believe you. I think you're lying about the car accident. You need to give me additional evidence before I believe that something that unlikely happened".

Ah, but maybe the people who are skeptical of the car accident are right? Maybe we should be more skeptical in general? It might be the polite thing to do to believe someone in such situations, but how do we know that that's actually the mathematically right thing to do?

Well, this is where the fact that this actually happened to me comes into play. I once got into a car accident on my way to a wedding. I was not hurt, nor was my car seriously damaged - but the whole affair did cause me to miss the entire wedding ceremony. I only managed to show up for the reception. That day, I told numerous people that I had gotten into a car accident, and gave it as my excuse for missing the ceremony. Not a single one of these people doubted me in the slightest: they all believed me. And they were right to do so, because I had in fact gotten into a car accident.

In fact, I've never heard of anyone, anywhere falsely using the "I had a car accident" excuse for missing an appointment. There are simply no reports of it that I know of. This is in spite of the fact that I have heard of numerous car accidents, and have been in one myself, and have heard it used as a genuine excuse before. All this, combined with the great deal of trust that the others correctly put in me when I told them of my car accident, tells me that the earlier 90% chance for the accident is too conservative. If I were to hazard a guess, I would say that such car accident stories are trustworthy about 99.9% of the time. That means that the posterior odds for the car accident are about 1e3, and the Bayes' factor from an earnest, personal testimony about a car accident is about 1e7 - although this is admittedly somewhat speculative.

So, if someone tells you about their car accident on a particular date, the Bayes' factor for their testimony should at least be 1e5 as a lower bound, and probably (but more speculatively) around 1e7.

Now, what if someone claims to have gotten into two car accidents in one particular day? The prior odds for such an event, assuming independence, is about 1e-7.6. Now, I have not heard anyone make this claim exactly, but I have heard of somewhat comparable events, like two tire blowouts happening on the same day (this, too, actually happened to me once). The comparison is difficult to make, as there are strong dependence factors and statistics on blowouts are harder to come by. However, going on my intuition, and my experience with similar events like blowouts, I would be willing to believe someone who claimed to have had two car accidents on a particular day, or at least give them even odds that they're not lying. This gives their testimony a Bayes' factor of about 1e8. While this is not a solid measure of the Bayes' factor on its own, it does validate my earlier estimation of the Bayes' factor being around 1e7.

Human death:

You're talking to a friend that you haven't seen in a year, and you're exchanging news about mutual acquaintances. You ask, "how's Emma doing?" Your friend then replies and says:

"Oh, you haven't heard? Emma... is dead. She was killed in a car accident. And you know how she was really close to her mom? Well, when her mom heard the news of Emma's death, she committed suicide - they say that they had the funeral ceremony for both of them together."

You may have guessed that this, too, actually happened to me. A friend of mine told me this tragic story about a girl we both knew. Don't be too concerned - the name of the girl has been changed, and this happened long ago - long enough ago that all the parties involved must have gotten well past the shock and the grief.

But, let us turn back to the question at hand. Should I trust my friend, on this very unlikely story? The yearly car accident fatality rate is about 1 per 10,000. The suicide rate is about the same. My friend's story, therefore, has a prior odds of about 1e-8 of being true. There is some dependence factors which increase the odds (a mother is more likely to commit suicide after her daughter's death), but the specifics of the story (the specific cause and timing of the suicide) would again decrease the odds. Let's say that they basically cancel each other out.

I'll go ahead and tell you that I did believe my friend. I did not really doubt his story. If I had to put down a number for my degree of belief, I would say that I gave his story about a 1e3 odds of being true. So the odds for this sequence of events went from a prior of 1e-8 to a posterior of 1e3, and therefore the Bayes' factor for my friend's testimony is about 1e11.

But was I right to trust my friend? Maybe I should have said back to him, "I don't believe you. Your story is just too ludicrous"? Well, as it turns out, I did get independent verification for a good chunk of this story later on. I really was right to trust my friend. Given that this is only a single instance of verification, this only validates that I was right to trust my friend, but not necessarily that I was correct to give the story a posterior odds of 1e3. So, at a minimum, I was definitely justified in giving my friend at least 1e8 for the Bayes' factor as a lower bound, and I feel that the correct value should actually be closer to 1e11.

So, here is the summary of the Bayes' factor evaluations thus far. Using publicly available statistics (car accident and fatality rates, suicide rates), and empirical events in my own life which I have personally experienced, lived through, and verified, I obtained two separate Bayes' factors for an earnest, personal testimony. In a story about a car accident on a given day, the lower bound on the Bayes' factor for that story should be 1e5, and the actual is probably closer to 1e7. In a tragic story about the unlikely death of a mutual acquaintance, the lower bound on the Bayes' factor for that story should be 1e8, and the actual value is probably closer to 1e11.

We see that in each case, even the minimum possible Bayes' factor exceeds 1e4. Recall that a Bayes' factor of 1e4 for an earnest, personal testimony would already put the probability of the resurrection at 99.999%. The more likely values we calculated in these specific cases, of 1e7 and 1e11, agrees very well with the value of 1e8 that I've used for the general case.

LinkedIn claim:

We are calculating empirical values for the Bayes' factor of a sincere, personal human testimony. Several lines of calculations have all converged around 1e8 as a typical value. In the last post, I gave some real-life examples that I have personally lived through and verified - and they validate the 1e8 value. But perhaps you're not convinced by the stories from my past. Fair enough - they're event that I have directly experienced, so they're empirical for me, but they're not empirical for you.

Here, then, is a calculation that anyone on the internet can verify to get an empirical value for the Bayes' factor of a human testimony.

Go on LinkedIn, and search for "PhD physics Harvard". You'll find many people who claim to be in the PhD program at Harvard University. You may need to upgrade your LinkedIn account to see the profiles for these people, if they're outside your network. Now, are these people telling the truth? And what ought we make of their claim that they're getting the most advanced degree in the most challenging field from the most prestigious university in the world? And what is the Bayes' factor for that claim?

To address this, we first need to find the prior probability for someone on LinkedIn being in the Harvard physics PhD program. For this, we'll need to gather up some numbers - all of which are readily available online.

First, let's get the number of people in Harvard's physics PhD program. This is easy enough - their department's webpage tells you that they have about 200 graduate students.

It's also easy to find the number of people on LinkedIn. Their website will tell you that they have more than 128 million registered members in the United States.

Now, we'll make the generous assumption that all 200 people in Harvard's physics PhD program are on LinkedIn. This means that the prior probability for someone on LinkedIn actually being in the program is about 200/128 million, or about 1e-6.

What about the posterior probability? Well, we can take the people on LinkedIn who claim to be in the Harvard physics PhD program, and actually investigate them one by one. Many research groups have their rosters published online, so you can easily find out whether someone really is in a physics research group at Harvard. You may also find their scientific publications or teaching records online, all of which can confirm their status in the program.

So, I searched on LinkedIn for "PhD physics Harvard". I spot checked more than a dozen people from the search results who claimed to be in the Harvard physics PhD program. I chose my sample over many pages across the unfiltered LinkedIn search results, so that the "relevance" of the search results to me will not influence my sampling.

What was the result? I found that every single person I checked was telling the truth. I could verify each of their claim independently from the LinkedIn page, nearly always from an official Harvard physics department page. Since I had checked over a dozen people, this represents a posterior odds of 1e1 at a minimum for these people really being in the Harvard physics PhD program.

This means that, at a minimum, the mere claim of these individuals on LinkedIn changed the odds for that claim, from a prior value of 1e-6 to a posterior value of 1e1. Therefore, the Bayes' factor for these claims have about 1e7 as a lower bound. The actual value is therefore well within range of the 1e8 value that we've been using.

It's also important to note how weak a claim on LinkedIn is compared to the kind of earnest, personal testimony that we're interested in. Anyone can get a LinkedIn account; they just have to sign up for it. They can then say whatever they want in that account. Furthermore, there is not much concrete negative consequences for lying, while the incentive of getting a job or a business contact can be quite appealing. But even with all this going against it, the people on LinkedIn turn out to be quite trustworthy, with the Bayes' factor for their claims having a value near 1e8.

The Bayes' factor for the disciples testifying to Christ's resurrection must be at least that much. Therefore, Christ almost certainly rose from the dead.

More evaluation for the Bayes' factor of a typical human testimony are coming next week.

9/11 false claims:

Here is yet another example from which we can empirically derive the Bayes' factor for a human testimony.

The September 11 terrorist attacks killed about 3000 people. It is the worst terrorist attack in world history to date. As such, it caused a great deal of shared grief and an outpouring of sympathy for the survivors and the families of its victims.

Of course, human being being what they are, some people falsely claimed that a close loved one had perished in the attacks. This got them a lot of sympathy - and more importantly, it got them a great deal of aid money, exceeding a hundreds of thousand of dollars in some cases.

This naturally leads us to ask - how reliable was a person's claim that they had lost a loved one in the 9/11 attacks? What was the Bayes' factor for such a claim? The numbers for this calculation are readily available. We just have to assemble them.

First, let's calculate the prior probability that someone really did lose a close loved one in the 9-11 attacks. We will assume that every one of the 3000 victims had about 4 loved ones (father, mother, sister, son, etc) whom we can consider "close", and that all of these loved ones lived in New York City. This gives 12,000, or about 1e4, close relation of the victims in a city with a population of 1e7. Therefore, the prior odds for a random person in New York City actually having a close loved one as a victim is about 1e-3.

Now, if someone claims that they had a close loved one die, what is the posterior odds that this person is actually telling the truth? One may assume that a vast majority of the 1e4 actual close relations of the victims made that claim. But how many false claims were mixed in with those? The specific number is not possible to determine (as someone could have lied so well that they were never suspected), but the article I previously linked mentions numbers like "dozens", "two dozen", or "37 arrests". Taking these numbers into account, let us be generous here and assume that there were 100, or 1e2, false claimants. The posterior odds are therefore 1e4:1e2, which is equal to 1e2.

Therefore, the Bayes' factor for someone claiming to have lost a loved one in the September 11th terrorist attacks is sufficient to take the odds from an empirically calculated prior value of 1e-3 to an empirically calculated posterior value of 1e2 - so it must be given a value of 1e5.

Nearly all of the numbers here are from Wikipedia or the New York Times. You can follow up on their sources and verify the values yourself. In the few places where I had to make assumptions, they have a definitive bias towards reducing the Bayes' factor - for example, the people who lost loved ones are not all confined to New York City, and 100 false claimants are a good deal more than two dozen. There's probably also a greater tendency for the truth-tellers to communicate their loss to more people in cases like these. Therefore, 1e5 is an underestimate of the true Bayes' factor. The actual value is greater - 1e6 seems like a reasonable guess.

Consider what this means: even when there was a clear reason to lie - that is, even when there was cold, hard cash at stake as a tangible reward for lying - people turned out to be fairly reliable overall. The Bayes' factor for their earnest claim about the personal tragedy of losing a loved one turned out to be about 1e6. Now, the general case would not have the explicit possibility of fraud as a precondition, and we would not be constrained to only consider the minimum value. Therefore a value of 1e8 for the general case is quite appropriate. That is a good estimate of the Bayes' factor for an earnest, insistent, personal testimony.

These Bayes' factor calculations will be summarized in the next post.

One in a million events happen every month:

(new material)

Other reports of a resurrection in history:

(part 30, maybe take out, merge with historical records section?)

Video of a lottery winner:

As one more confirmation of that 1e8 number, take a look at this video - it show's a woman's reaction to an acquaintance claiming to have won the lottery. Now, did that woman seem like a gullible idiot to you? I didn't feel that way. She starts off quite skeptical, but not dismissively skeptical. You can then see the man's sincerity working on her. Her degree of belief is clearly somewhat close to even odds right before the numbers are confirmed. I think her overall reaction is pretty rational. Now, there are some small differences between the video and my examples. For instance, she knows that there's a winner out there, and the man making the claim is already an acquaintance - but on the other hand, this result is achieved with little effort on the man's part, taking only minutes of insistence. The man being an acquaintance also reduces the "licenses plate effect". On the whole, you can see her mind being pulled through a Bayes' factor of something like 1e6 within mere minutes, in good accord with rationality, in a situation pretty similar to what I described in my examples. So 1e8 for something like the disciple's testimony about the resurrection is quite reasonable, and remains the best value to use.

revisiting our initial calculation.


Conclusion: The strength of a human testimony is firmly established and understood.

Let us summarize our investigation into the Bayes' factor for a human testimony.

At the beginning of this series, we began by examining our gut feelings on how much credit we would give to someone who claims to have won the lottery or been struck by lightning. From this initial calculation, using just some intuition, we got a variety of numbers for the Bayes' factor, ranging around 1e7 to 1e9. The number we ended up using, 1e8, started from these calculations.

That's a good start, but it needs empirical backing. The first natural experiment we used to verify this number was the case of the people who lied about winning the 1.6 billion dollar Powerball lottery. The result from this calculation was about as good as it could possibly be expected; 1e8 really turned out to be the correct order of magnitude for the Bayes' factor, when someone claimed that they had they had won the lottery.

We then investigated the case of someone missing an appointment due to a car accident. The claim of a car accident on a specific day turned out to have a Bayes' factor of 1e5 as a lower bound, while its true value was estimated to be around 1e7.

We next investigated the tragic story of a young woman dying in a car accident, and her mother committing suicide when she heard the news. The testimony of the person who related this story was calculated to have a Bayes' factor of 1e8 as a lower bound, while its true value was estimated to be around 1e11.

For the claims of being in Harvard's physics PhD program, the Bayes' factor was found to have a lower bound of 1e7, with no estimate for the most likely value. And for the case of people claiming to have lost a close loved one in the 9/11 attacks, the Bayes' factor turned out to be about 1e6, despite the fact that there was cold, hard cash to be won as a strong temptation to lie.

So, the Bayes' factor for an earnest, sincere, insistent personal testimony really is about 1e8, and this is born out by multiple lines of thought, and verified by multiple cases of empirical inquiry.

It is important to note that these examples were merely the first ones that came to my mind which I could also get good numbers for. There is no selection bias here. There is not a set of other examples which I chose not to use because they did not prove my point or suit my purpose. In fact, I encourage you to come up with your own examples through which you can compute the Bayes' factor of a human testimony. Compare your answer with mine, and independently verify my values.

It is also important to acknowledge that there is variance in the Bayes' factors. That 1e8 is a typical value, and it will naturally change when we put conditions on it. For example, the relatively low value of 1e6 for people claiming to have lost loved ones in the 9/11 attack can be attributed to the possibility of dishonest gain through fraud. On the other hand, the high value of 1e11 was obtained for a friend telling me an unlikely story, and its greater Bayes' factor can perhaps be attributed to the fact that I was friends with this person. It seems that such considerations can shift the Bayes' factor by a couple orders of magnitude, although it's also worth noting that there are also cases like reporting the moves of a chess game, where the Bayes' factor exceeds 1e120.

Remember, a Bayes' factor of 1e4 (and its corresponding discount for lesser testimonies) still gives a 99.999% chance that Jesus really rose from the dead, even starting from the ridiculously low prior odds of 1e-22. So for our purposes, the important thing in these investigations is that the Bayes' factor always exceeded 1e4 - even for the minimum estimates in each case. Furthermore, the minimums always exceeded 1e4 by more than a couple of orders of magnitude - which is the above-mentioned typical variation in the Bayes' factor under different conditions. Meaning, EVEN IF you believe that the disciples had a good reason to be deceptive or delusional, there's STILL enough evidence in their weakened testimonies to conclude that Jesus did really rise from the dead. That's how strong the case for the resurrection is.

Of course, we've already covered the issue of how the disciple's testimonies may vary from the "typical" testimony. We've seen that in every way, their testimonies are in fact stronger than the testimonies of a "typical" person. The variation in their Bayes' factor from 1e8 is therefore going to be towards larger numbers, like 1e11 or 1e120, rather than towards 1e4. 1e8 is an underestimate of the Bayes' factor, and therefore 1e32 is also an underestimate for the odds for Christ's resurrection. Jesus almost certainly rose from the dead.

(add the additional examples. One in a million, video of lottery winner, etc.)

In the next post, we'll tackle this question of the Bayes' factor for the resurrection testimonies from a different angle.

Chapter 3:
Double checking our conclusion against other reports of resurrections in history

Rational: Are there other historical cases of reported resurrections? How good are they?

So, our previous Bayesian analysis of the resurrection compels us to believe that Jesus really rose from the dead. But, as an additional layer of verification, let's approach the problem from a slightly different angle, and see if we come to the same conclusion.

In our Bayesian analysis, the odds for Jesus's resurrection went from a prior value of 1e-22 to a posterior value of 1e32 - meaning, the Bayes' factor for the testimonies in 1 Corinthians 15 was about 1e54. Another way of stating that is to say that the evidence of those testimonies is 1e54 times better explained by an actual resurrection than by naturalistic alternatives.

Now, if you want to cling to a naturalistic alternative, you must believe that this Bayes' factor value is incorrect. That it is not really that large. That the true value is insufficient to overcome the small prior odds. That a naturalistic alternative can sufficiently explain the evidence, so as to make an actual resurrection unnecessary.

Well, can you demonstrate that empirically?

If naturalism can sufficiently explain the evidence for Jesus's resurrection, I expect there to be some non-resurrection cases where the same level of evidence was achieved through ordinary means - through naturalistic chance, as it were. It would be a strange naturalistic explanation indeed that works only once for the specific case that we're trying to explain, and never works again.

Here's what I mean: let's say that you think the resurrection testimonies are totally worthless and changes nothing about the probability that Jesus rose from the dead. This would correspond to a Bayes' factor of 1, meaning that a non-resurrection is equally likely to produce these testimonies as a genuine resurrection. Well, in that case, you ought to be able to produce literally billions of cases throughout history where random people are said to have been resurrected, with each of these cases having the same level of evidence that the New Testament has for the resurrection of Jesus. Can you produce these cases?

Say that you're willing to be slightly more reasonable: you think the Bayes' factor for the resurrection testimonies is 1e6 - far smaller than 1e54, but still significantly greater than 1. Effectively, you believe that the testimonies clearly do count as evidence, but that it's just not enough to overcome the prior. Well, a Bayes' factor of 1e6 corresponds to saying that a non-resurrection still has one millionth the chance of producing New-Testament level of testimonies compared to a genuine resurrection. Again, given that there have been billions of people who died throughout human history, this means that you should still be able to produce thousands of accounts of someone rising from the dead, with each account having as much evidence as the New Testament has for Jesus's resurrection.

You can easily do the same calculation for a Bayes' factor of 1e9. Following the examples above, If you think that the Bayes' factor is only that large, then you should still be able to find at least one other case where a natural death and no resurrection still produces the same level of evidence as the New Testament has for Jesus's resurrection.

Ah, but what if you believe, as I do, that the Bayes' factor is at least 1e54? Wouldn't that require observing less than one case of something? How does that work out? Well, 1e54 is 1e9 raised to the 6th power. So, if 1e54 is really the Bayes' factor, you'd expect that the nearest case of a non-Christian resurrection story to have about one-sixth the evidence that Jesus's resurrection has. That is how we can at least coarsely verify the Bayes' factor of 1e54.

Think of the process in this way: say that there's a record of a million coin flips. While examining that record, I come across a sequence of 10 heads in a row, and say "Wow, that's amazing! These coin flips couldn't have been random!" Now, if you wanted to debunk me by showing that random chance can easily produce such sequences, you can say "Actually, the chances of getting 10 heads in a row randomly is only 1 / 2^10, or about 1e-3. The Bayes' factor of your sequence for your hypothesis is therefore only 1e3. In a million coin flips, you'd expect to see something like this about a thousand times". You can then proceed to point out those thousand other "10-heads-in-a-row" sequences in the coin flip record, and that would validate your Bayes' factor estimation.

However, let's say that I then come across a sequence of 60 heads in a row. I say again, "Wow, that's amazing! These coin flips are clearly non-random! I think the chances of a sequence like this is 1 in 1e18". How could I empirically prove that my estimate is correct, when the probability is so small? Wouldn't I naturally expect zero such "60-heads-in-a-row" sequences from a million flips?

It's simple. Just find the sequence with the longest chain of heads in the coin flip record. In a million flips, you'll probably see a maximum sequence with about 20 heads in a row, which has about a one in a million chance to occur. This means that a 40-head sequence will happen once in a million-squared coin flips, and a 60-heads-in-a-row will happen once in a million-cubed (or 1e18) coin flips. Thus, by verifying that the longest sequence of heads has about 20 head in it, I also verify that the chances of 60 heads in a row is about 1e18. So even when the Bayes' factor is extremely large for a very strong piece of evidence, you can still get an estimate for that Bayes' factor by seeing what fraction of that evidence is duplicated by chance in the population at large.

I'm making some simplifying assumptions here, such as independence of events and a somewhat "reasonable" distribution over the level of evidence. As you'll see, the case for the resurrection will again turn out to be so strong that some small amount of sloppiness like this will simply not matter.

So, it basically comes down to this: you think that the evidence for the resurrection isn't good enough? Well then, start citing other, non-Christian examples in history where someone comes back from the dead. We'll then see how the best of these measure up against the evidence for Christ's resurrection, and then see how the Bayes' factor calculated this way compares to our previously calculated value.

We'll begin this calculation next week.

The conditions: the requirements for "matching" a testimony.

We are interested in quantifying the Bayes' factor for the testimonies concerning Christ's resurrection. We will do so by comparing the level of evidence for Christ's resurrection against the level of evidence we find in world history through a naturalistic process and outlook. Here is how the procedure will work:

As a conservative estimate, let us say that there have been 1e9 reportable, naturalistic deaths throughout world history. Then, a Bayes' factor of 1 would correspond to finding that all 1e9 of those deaths also resulted in resurrection stories, each as well-evidenced as Christ's resurrection.

A Bayes' factor of 1e9 would mean that there is still one other naturalistic death which resulted in a story on par with Christ's resurrection.

A Bayes' factor of 1e18 would mean that in all the rest of the world, among the 1e9 reportable, naturalistic deaths, there may be some deaths which has a resurrection story - but they would only have at most about half the evidence that Christ's resurrection has. The rationale is that if achieving half-evidence has a 1e-9 chance of happening, then the chance of that happening twice for the same event would take 1e9 squared, or 1e18, number of events.

By the same reasoning, a Bayes' factor of 1e54 would imply that the closest event to Christ's resurrection would have about 1/6th the evidence that Jesus's resurrection has.

So then, what are these evidences for Christ's resurrection? And what would 1/6th of that evidence look like? For our current purposes, we're using the testimonies enumerated in 1 Corinthians 15 - that's the passage from which we calculated our value of 1e54 for the Bayes' factor. In getting that number, I specifically accounted for the earnest, insistent, sincere personal testimonies of Peter, James, and Paul, as well as the group testimonies of the other apostles and the 500 disciples.

That's five distinct sets of testimonies. Not all of them were given equal weight in my calculations - the least of these were only responsible for 1e8 out of 1e54 in terms of Bayes' factors, or about 1/7 of the total weight of evidence. So, let's be conservative here (as I've always been throughout this entire series), and say that if any other non-Christian story about a resurrection matches any one of the five sets of testimonies in Acts 15, that would count as achieving 1/6th the evidence of Christ's resurrection.

What would it take to "match" one of these sets of testimonies? Well, the idea here is that the overall quality of the evidence should be on par with the evidence we have from the testimonies mentioned in 1 Corinthians 15. So:

To match Peter, James, or Paul's testimonies, we will require an earnest, insistent, and personal testimony by a single named individual, whom we can historically locate with great precision. We will also require that a good amount of information is available about the life of the person giving the testimony.

To match the testimonies of the other apostles would require an earnest, insistent, and personal testimony by a group of people, most of whom are named and known in history, who can be located with good precision.

To match the testimonies of the group of 500 disciples would require the earnest personal testimonies from a large, specific group of people, who can be well-located in history. They don't have to be named, and they don't have to be insistent in their testimony. But they should be well-defined enough that many of them could be individually pointed out by a well-known historical personage like Paul.

That will be our methodology. Starting in the next post, we will examine the claims where someone was said to have been raised from the dead, like Jesus. We will look at these claims and assign numerical values to the amount of evidence they have, using the criteria specified above. We will then assign a Bayes' factor for the evidence for Jesus's resurrection, based on how it measures up against the evidence for these other supposed resurrections.

The other historical records:

Apollonius of Tyana:

Fortunately, skeptics of Christ's resurrection often do some of the early leg work for us, in that they compile lists of purported people who have been said to be like Christ for one reason or another. We'll look at a representative sample from such lists.

First, let us consider Apollonius of Tyana, who is sometimes compared to Christ because they were both philosopher/preachers in first century Rome, to whom miraculous powers are attributed. Wikipedia has a list of similarities between Jesus and Apollonius, which includes a wondrous birth, the ability to heal the sick and raise the dead, a condemnation by Rome, and an ascension into heaven. That sounds pretty similar, no? So how does the evidence for Apollonius's "resurrection" hold up?

Pathetically. Most of the information on Apollonius comes from Philostratus, who was paid to write a biography of Apollonius well over a hundred years after Apollonius's death, and after Christianity was already a thing. This biography only "implies" that Apollonius underwent heavenly assumption. Furthermore, the chief primary source for this biography is one Damis, a disciple of Apollonius, who is unknown outside of this biography. And to top it off, Philostratus specifically writes that Damis had not recorded anything about Apollonius's death. The stories of his death and supposed heavenly assumption are in a part of the biography that are filled with 'some say this, some say that' stories, which, by the author's own admission, he wrote because he felt that his story needed to have a natural ending.

So, the evidence for Apollonius's "resurrection" comes down to one author, who wrote more than a hundred years after the event, who says that he's getting his information second-hand from a Damis that nobody else has heard of, who then says that the "resurrection" bit - which is only implied - doesn't even come from Damis.

Compare that to the evidence for Christ's resurrection, in the form of the testimony of his disciples. 1 Corinthians 15 was written within a couple decades of the event, and it contains a creed that was formulated mere years after the resurrection. We have the personal, first-hand testimonies of the people who have seen the risen Christ. They clearly say that it really happened, and that it transformed their own lives. Each of these disciples appear in multiple other sources, and bear the same witness about Christ's resurrection in those sources.

So... now I'm suppose to compare the strength of the evidence between these two? Well, let's see. Remember our previous criteria, about what it would take to "match" 1/6th of the evidence in 1 Corinthians 15. Can we say that maybe that Damis's testimony about Apollonius's resurrection matches the testimony of Peter, James, or Paul? Well, no. Damis never made that testimony, nor is he anything like those three individuals on the quality of historical information we have on him. So then, all that's left as evidence is "some say that Apollonius rose from the dead", stated more than a hundred years after the fact?

That is essentially no evidence. But since I have to give a numerical estimate, I would be generous and say that Damis's "testimony" counts as an order of magnitude less than that of Peter, James, or Paul. I will also generously grant the "some say..." part of the story as being a order of magnitude less than that of the 500 witnesses that Paul mentions. So, that comes to:

1/6 (matching a single element in 1 Corinthians 15)
× 1/10 (an order of magnitude less)
× 2 (two such instances),
= about 1/30th of the evidence that we have for Christ's resurrection.

We will continue with other personages in the next post.


Next, let us consider Zalmoxis, whom Herodotus writes about in his "Histories" as a divinity in the religion of the Getae. Herodotus wrote that Zalmoxis's followers believed they have a form of immortality in him, and performed a kind of human sacrifice to communicate with him through death.

According to Herodotus, he was told by certain non-Getae peoples that Zalmoxis was really a man - that he was teaching his countrymen some philosophy, but then hid himself in a secret underground housing for three years while people thought he was dead. He then came back out and showed himself alive, and this caused the people to believe his teachings.

And... that's it. That's the substance of this Zalmoxis and his "resurrection". Apparently this is one of the best examples that the world can come up with when asked about non-Christian resurrection stories. And yes, some people really have tried to link this "resurrection" to Jesus's resurrection, in an attempt to discredit Christianity. This, in spite of the record having no witnesses testimonies of any kind, nor even a group of people who can clearly be said to believe that someone came back from the dead.

Again, using the metric derived from 1 Corinthians 15, how does this measure up against the evidence for Christ's resurrection? Is anything about Zalmoxis's "resurrection" comparable with the testimonies of Peter, James, or Paul? Well, no. Zalmoxis has no witness testimonies, period - let alone any named witnesses among historically known persons. This means that nothing about Zalmoxis is comparable to the testimony of the apostles as a group, either. At the end of the day, all the evidence for Zalmoxis's "resurrection" comes down to "some people might have said that a god, who might have been a real person, might have come back from the dead". Note that all the "might have"s in that sentence are part of the historical evidence. It is not an external skeptic injecting doubt into the story, it's actually how the story is handed down to us through history.

So... I would again say this is pretty much no evidence. But again, because we need a quantitative value, I will be generous and say that this is an order of magnitude less than the evidence of the 500 witnesses in 1 Corinthians 15. That gives Zalmoxis 1/6 * 1/10 = 1/60th of the evidence that we have for Christ's resurrection.


Let's next look at Aristeas, who is another character in Herodotus's "Histories". He is said to have been a poet. The "Histories" relate how Aristeas "suddenly dropt down dead" one day (in front of just one witness), but then his body could not be found and he was seen alive - once close to the time of his death, and then seven years later, when he appeared in another town and wrote a poem.

Here's the thing about this story: it was already at least 240 years old when Herodotus was telling it. Then, Herodotus says that some people say that Aristeas appeared again (as a "ghost" or an "apparition") after those 240 years, and instructed these people to build an alter to Apollo and a statue of Aristeas himself.

Again, that's about it. The whole story only takes up a couple of paragraphs in Herodotus's "Histories". Now, it's not quite clear that a "resurrection" had taken place - the first part of the story sounds more like a fainting or a disappearance, and the second one is called a "ghost" or an "apparition" by the people who were suppose to have seen it, who presumably had no means of personally identifying Aristeas. But let's ignore that for now. What kind of evidence - what kind of witness testimony - do we have for this story, and how does it compare to the story of Christ's resurrection?

Well, once again we have no named witnesses. The first part of the story is at least 240 years old at the time of the telling - so no witnesses, of any kind, are even possible. The second part, where a ghost or an apparition instructs people to build an alter and a statue, may be a bit more credible. We seem to at least have a group of people who were instructed to build a specific alter and statue, and Herodotus might have conceivably met the individuals who claimed to have personally received these instructions. On the other hand, they are never identified more specifically than "the people of Metapontion", and it's unclear whether this is simply a story that the Metapontines told about their alter and statue. Furthermore, it's not even clear how long ago this was supposed to have happened - the story about the apparition might as well have happened another 240 years ago from the time that Herodotus relates the story, judging from the scant details.

So, once again the testimony evidence here only turns out to be of the "some people say..." kind. The closest thing we can relate this to is the testimony of the 500 witnesses in 1 Corinthians 15. Given the rather shadowy nature of the "apparition", and the uncertainty about whether Herodotus has any specific primary witnesses in mind, I would generously say that this counts for maybe a fourth of the evidence of the 500 witnesses. So, Aristeas's "resurrection" has about 1/6 * 1/4 = 1/24th of the evidence we have for Christ's resurrection.

We'll present some more examples next week.


How about we look at some ancient gods? Jesus is often compared to the gods in other religions, but can any of them actually serve in our comparison of historical evidence for a resurrection?

Mithra, for instance, is a god in the Persian religion of Zoroastrianism, who then inspired a Roman mystery religion. He often appears on lists of gods that Jesus was supposed to have been copied from. But... um... it seems that he was never actually said to have been a human, or any kind of a historical figure, in either the Persian or the Roman variants. He doesn't even die, let alone rise from the dead, even in his mythologies. Furthermore, any specific details or even general plot points is notoriously difficult to extract from any Mithra mythology. The Roman version of Mithra was worshiped in a mystery religion, and none of their written narratives or theology survive - we only have some iconography to glean what we can of this Mithra. In the Persian version, Mithra is mentioned in some hymns (Yashts), which are again very short on details, mythology, or narrative. In all cases, he is always presented as a mythic entity, and the scant stories about him are always framed in that context.

So, on his comparison with Jesus, Mithra fails on this crucial point of historical existence. For our purposes, this also means that we can safely say that there is no evidence for Mithra's resurrection. Indeed such a claim is never even made, or even dreamt of - someone would first have to claim that Mithra was a historical figure.


(part 21, new material)


Horus, an ancient Egyptian god, along with his father Osiris, are some more gods who are sometimes compared to Jesus - and they, too, fail the "historical existence" test. As with Mithra, all of the stories concerning these gods take place on a purely mythological level, and there are no claims to them having been a real, historical figure. For our purposes, it's clear that their story presents no evidence for a historical resurrection. But at least Osiris has a mythological story where he comes back from the dead. Of course, it's not even clear that there was ever a group of people who might have claimed to have been historical witnesses to this - all ancient sources (Pyramid Texts, Palermo Stone, etc.) which mention this story always present it something that took place a long time ago, in an mythic age.

So, in assigning a level of evidence to this, we'll be extremely generous and again count this as an order of magnitude less than the evidence of the 500 witnesses in 1 Corinthians 15. Recall that this comes to 1/6 * 1/10 = 1/60th of the evidence that we have for Christ's resurrection.


Dionysus is another god, this time from the Greek pantheon, who is superficially compared to Jesus but fails the "historical existence" test. Yes, there is a mythological story where he is killed as an infant then re-incubated in Zeus's thigh - but none of the sources that mention this mythology pretends to be history. Dionysus's situation with regards to his "resurrection" is therefore similar to that of Osiris - there is virtually no historical evidence for his "resurrection".

As with Osiris, we'll again be extremely generous and rate him as having 1/60th of the evidence for Christ's resurrection.

We will discuss Krishna in the next post.


We now come to Hinduism's Krishna, who's another god that's sometimes compared with Jesus. He's said to be have been the incarnation of Vishnu, who is either the supreme god, or one of three or five most important gods, depending on the specific tradition in Hinduism.

Krishna has perhaps a greater claim to a real, historical substance compared to the other gods we've covered. For starters, he is at least said to have been born as a human. He is said to have gotten married and ruled kingdoms and fought battles. There is a great deal that is said about Krishna - but we are, of course, primarily interested in the story of his death and "resurrection".

The main literary sources we have on this part of Krishna's life are the Mahabharata and the Srimad Bhagavatam. They tell the story of how Krishna, at the end of a long and eventful life, intended to leave the world. He was then shot by a hunter named Jara, with an arrow through the foot. This marked the end of Krishna's life, for thereafter he immediately ascended to go to his own abode, leaving earth.

So, what are we to make of this "resurrection" story? What kind of evidence is there for it? Let us first try to establish the setting. These stories take place in ancient India, and Krishna is proposed to have lived some time between 3200 and 3100 BC, although there are some wildly differing estimates. These are quite large uncertainties, from a very long time ago - right at the edge of pre-history. These issues, by themselves, might not cause too much concern - until we attempt to date the writing of the Mahabharata, which contains these stories.

Dating the Mahabharata is tricky - it is a massive work, composed of multiple layers. Current scholarship estimates that the oldest layers are from around 400 BC, and the origin of the stories within it can perhaps be extended back to 1000 BC. In other words, the stories of Krishna were, at best, already thousands of years old at the time that they were recorded. Therefore, no personal, firsthand testimony to Krishna's death and ascension are possible in this work.

Okay - but what if we ignore the scholarship, and and go with the Hindu tradition which says that the Mahabharata was authored by the legendary sage Vyasa? Unfortunately, this doesn't help things at all. We know little about a historical Vyasa. When did he live? When did he write? We can no more anchor him in history than we can Krishna.

Complicating matters further is the story structure of the Mahabharata. You see, the death and ascension of Krishna is not just told as a story; it is framed as a story being told by Vaisampayana (a student of Vyasa) to the king Janamejaya (supposedly a great-grandson of a character in the Mahabharata), many years after the fact. But that's not the end of it - this story is further framed as a story being told by Ugrasrava Sauti, even more years later. So, the story of Krishna's ascension is a story (about Krishna), within a story (being told by Vaisampayana), within a story (being told by Ugrasrava Sauti), within a work (the Mahabharata itself, which was presumably written down some time afterwards). All this "story-within-a-story" structure sounds like a device for saying "once upon a time...", and makes the story sound like something told about "a friend of a friend". But let us ignore that for now. Even if we were to take the Mahabharata entirely at face value - an outlandishly generous acquiescence - we would still be forced to conclude that this story was already incredibly old at the time of the recording, and its content disqualifies itself from being considered a primary account, due to its story-within-a-story structure. Again, no personal testimonies are possible.

But - what if the dates for Krishna's life are mistaken? What if he lived more recently than in the 4rd millennium BC, and the portion of the Mahabharata which contains his ascension were written closer to the actual event, and the rest of the Mahabharata, including the story-within-story structure, was built up later? Well, that's a lot of "what-if's" - and while that does get the text closer to the event, it's still of no help in solidly placing Krishna in history, or producing personal testimonies from any witnesses to his ascension.

Going to the Srimad Bhagavatam instead of the Mahabharata doesn't help here - for the Srimad Bhagavatam was written even more recently than the Mahabharata. Modern scholarship places its composition as some time between 500 to 1000 AD, and it references parts of the Mahabharata. In fact, its other name - Bhagavata Purana - means "Ancient Tales of Followers of the Lord". The work itself acknowledges that these are "ancient tales", right there in the title. It cannot possibly produce the kind of testimonies we're looking for.

Let's compare all this to the evidence for Jesus's resurrection. Even if we only consider those modern scholars that are skeptical and unbelieving, the New Testament was mostly completed within decades of Christ's death and resurrection. 1 Corinthians, from which we got the summary of the evidence for Christ's resurrection, was written a mere 20 years after the event. The creed within it comes within several years of the event itself. Furthermore, we have numerous records within the New Testament of people claiming to have personally seen the risen Christ. Multiple such claims are in fact made in the first person within the New Testament text itself.

These are stark differences compared to the ascension of Krishna. We have time gaps of years compared to millennia, and personal, firsthand testimonies instead of a story about a story about a story about an ascension. It may be that Krishna was a real person who once lived a remarkable life. It may be that the Kurukshetra War actually took place. But in judging the amount of evidence for Krishna's ascension, there can be no real comparison to the evidence for Christ's resurrection.

But in the end, we still need a numerical value for the level of evidence for Krishna's ascension. Well, we can certainly say that some people say that Krishna "rose from the dead". But we cannot historically locate any group of people who first personally testified to this fact, like we can with the 500 witnesses in 1 Corinthians 15. Nor can we find any group of witnesses corresponding to the apostles, or to the specific named witnesses in 1 Corinthians 15. In the end, we just seem to have the story in the Mahabharata, with the version of the story in Srimad Bhagavatam being a later telling of the same story. Previously, I've assigned such "some people say" stories 1/10th of the level of evidence of the 500 witnesses. But given the sheer size of the works about Krishna, I'll increase this to 1/4th of the level of evidence of the 500 witnesses. That means that the evidence for Krishna's ascension amounts to 1/4 * 1/6 = 1/24th of the evidence for Jesus's resurrection.

There's more cases to consider in the next post.


Let us now turn to some figures from Buddhism who are said to have appeared after their deaths.

Bodhidharma is the Buddhist monk credited with bringing Chan Buddhism to China, some time around the 5th century AD. Here is Wikipedia's summary of the legend surrounding his death:
Three years after Bodhidharma's death, Ambassador Sòngyún of northern Wei is said to have seen him walking while holding a shoe at the Pamir Heights. Sòngyún asked Bodhidharma where he was going, to which Bodhidharma replied "I am going home". When asked why he was holding his shoe, Bodhidharma answered "You will know when you reach Shaolin monastery. Don't mention that you saw me or you will meet with disaster". After arriving at the palace, Sòngyún told the emperor that he met Bodhidharma on the way. The emperor said Bodhidharma was already dead and buried and had Sòngyún arrested for lying. At Shaolin Monastery, the monks informed them that Bodhidharma was dead and had been buried in a hill behind the temple. The grave was exhumed and was found to contain a single shoe. The monks then said "Master has gone back home" and prostrated three times: "For nine years he had remained and nobody knew him; Carrying a shoe in hand he went home quietly, without ceremony."
So, that's something. We not only have the usual "group of people who believe" that Bodhidharma rose from the dead, but also a named figure, one "Ambassador Sòngyún of northern Wei", who at least sound like a historical person. So, how should we evaluate this story?

As before, we first ask where this story comes from. It turns out that the source for this story is the Anthology of the Patriarchal Hall, which was compiled in 952 - about 400 years after Bodhidharma is supposed to have died. Again, this is far outside a human lifetime, and that makes it impossible to find the kind of personal testimonies of historical individuals that we're looking for.

As for "Ambassador Sòngyún of northern Wei" - well, it turns out that he really was a historical person - a Buddhist monk who was sent into India to acquire some Buddhist texts, some time around 520. But this does not really help the case for Bodhidharma's "resurrection", because none of the texts that mention Song Yun or his journey mentions this "resurrection". The event therefore seems to be a later, legendary addition.

The other sources on Bodhidharma, many of which are earlier than the Anthology of the Patriarchal Hall, also force us to draw the same conclusion. None of them mention this story of Bodhidharma "going home". It is clearly a later, legendary addition, and Wikipedia has no qualms about labeling it as such.

Let's now assign a numerical value to the level of evidence in this story. There's the usual "some people say this happened" dimension to the story, which again gets counted for 1/10th of the 500 witnesses in 1 Corinthians 15 - that is, as 1/10th of 1/6th of the evidence for Jesus's resurrection. As for "Ambassador Sòngyún of northern Wei", having a named, real, historical witness would count as a full 1/6th of the evidence, except that this witness is named 400 years after the fact. This would still count as some non-negligible fraction of that 1/6th, and we'd have to add that in.

But nearly all of this gets wiped out by the strong evidence against the story from the lack of mention in the earlier sources, indicating that this whole story is a later, legendary addition. In the end, the level of evidence for Bodhidharma's "resurrection" can't amount to more than the "some people say" level - the usual 1/60th of the amount of evidence for Christ's resurrection.

Next week we'll examine another Buddhist monk.


Puhua (known as Fuke in Japan) was a Chinese Buddhist monk, who supposedly lived around 800AD. He, too, is said to have not really died. He may or may not have been a real individual. If real, he was a student of Linji (known as Rinzai in Japan), who was another Chinese Buddhist monk, who founded the Linji school of Chan Buddhism.

Here's the story of Puhua/Fuke's death and "resurrection" as told in the Record of Linji, quoted by Wikipedia:
"One day at the street market Fuke was begging all and sundry to give him a robe. Everybody offered him one, but he did not want any of them. The master [Linji] made the superior buy a coffin, and when Fuke returned, said to him: "There, I had this robe made for you." Fuke shouldered the coffin, and went back to the street market, calling loudly: "Rinzai had this robe made for me! I am off to the East Gate to enter transformation" (to die)." The people of the market crowded after him, eager to look. Fuke said: "No, not today. Tomorrow, I shall go to the South Gate to enter transformation." And so for three days. Nobody believed it any longer. On the fourth day, and now without any spectators, Fuke went alone outside the city walls, and laid himself into the coffin. He asked a traveler who chanced by to nail down the lid. The news spread at once, and the people of the market rushed there. On opening the coffin, they found that the body had vanished, but from high up in the sky they heard the ring of his hand bell."
As before, we want to evaluate the evidence for this story, and begin by inquiring about the source of the story.
We've said that this story comes to us through the Record of Linji - a work that was not consolidated until more than 250 years after Linji's death in 866. Puhua, if he was real, died before Linji - as the story itself makes clear. Therefore, this story about Puhua's death and "resurrection" was recorded more than 250 years after the event itself. Again, the large gap, which far exceeds a human lifetime, makes it impossible for us to find anything like the personal testimonies of historical individuals.

More damning still is the other, earlier account of Puhua's death, in the Anthology of the Patriarchal Hall - the same Anthology that recorded Bodhidharma's "resurrection". This text is also known as the Zutang ji, and it contains the first mention of Linji as well as telling the following story of Puhua's death (look on p.312. "ZJ" refers to Zutang ji):
One day Puhua, carrying an armload of coffin-planks, went about town bidding farewell to the townspeople, saying, “I’m leaving this life.” People gathered in crowds and followed him out of the east gate. He then said, “No, not today!” The second day he went to the south gate and the third day to the west gate. By that time fewer people were following him, and not many believed him. On the fourth day he went out of the north gate, but no one followed him. He dug a tunnel, lined it with bricks, and died therein.
This is, of course, essentially the same story as the one found in the Record of Linji - except there is no resurrection. So, Puhua died, supposedly in 840 or 860. We then have the Anthology of the Patriarchal Hall, written in 952, which mentions Puhua's death but says nothing about a vanished body or a resurrection. We then finally come to the Records of Linji, which was consolidated after 1100, where a resurrection shows up attached to the end of the same story as the one in the Anthology of the Patriarchal Hall. We furthermore know that the Anthology of the Patriarchal Hall is not shy about putting in resurrection stories, since it included one for Bodhidharma. So, why does it not include Puhua's resurrection story? Because the story did not exist yet. The obvious conclusion is that Puhua's "resurrection" is a legend developed after 952.

Again, it's difficult to compare something like this to the evidence for Jesus's resurrection in the New Testament. None of the New Testament makes any sense without Jesus having risen from the dead. The whole corpus, from beginning to end, testifies to Christ's resurrection, without ever wavering from that truth. But, we're suppose to assign a comparative numerical value to the level of evidence for Puhua's resurrection - so the only thing we can do is to generously give it the "some people say" value of 1/60th of the evidence for Christ's resurrection.

The next post will begin a short intermission, where we'll discuss the series thus far.

Conclusion: Our previous calculation is fully validated

So, let us summarized these non-Christian accounts of a resurrection. For each supposedly "resurrected" person, the following table shows the level of evidence associated with their resurrection account, expressed as a fraction of the evidence we have for Christ's resurrection:

Name of the personThe level of evidence
Apollonius of Tyana1/30th

Here's how this looks like in a histogram:

What does all this tell us? Quite a bit - we'll discuss that starting next week.

Let us recall our purpose in collecting these non-Christian stories about a "resurrection": we wanted to verify our Bayes' factor for the evidence of Christ's resurrection. My claim is that it's at least 1e54.

The first part of our plan was to find the non-Christian resurrection story with the most evidence behind it. If we make the naturalistic assumption about these stories, we can then say that this level of evidence is approximately what corresponds to a Bayes' factor of 1e9. For by the virtue of having the most evidence, such a resurrection story would have narrowed the field down to itself - one case - from the approximately 1e9 reportable deaths in history.

As it turned out, the "resurrection"s of Krishna and Aristeas had the most evidence behind them, amounting to roughly 1/24th of the evidence for Christ's resurrection. According to our program, this must be assigned a Bayes' factor of roughly 1e9. Then 24 times that amount of evidence would correspond to raising the Bayes' factor to the 24th power - meaning, the evidence for Christ's resurrection has a Bayes' factor of... 1e216.

So yes, that does verify that the Bayes' factor is "at least 1e54". It furthermore demonstrates how much of an underestimate that value is. Recall that, in a slightly different context, I mentioned that the full odds for the resurrection would be far in excess of 1e100, and that our values for the Bayes' factors were drastic underestimates. All that is verified by this completely different methodology, of comparing with non-Christian resurrection stories.

But that's not all. This comparison also provides yet another layer of verification, in that it allows us to check the Bayes' factor of 1e8 for a disciple's testimony about Christ's resurrection. You see, among the non-Christian resurrection stories we've seen, there was not a single case of a person making an earnest, insistent testimony about someone rising from the dead. That says something about the strength and rarity of such testimonies. Granted, we have not investigated every existing non-Christian resurrection story - but if such a testimony really has a Bayes' factor of 1e8, there should be about ten such testimonies for us to find. The fact that we have not found a single one puts a lower bound on the Bayes' factor, of just about 1e8. As usual, there's some nitpicking possible, depending on whether you think there are a hundred or thousands of non-Christian resurrection stories. But it's unlikely for any of that to change the value of 1e8 by more than a couple of orders of magnitude. So our estimate about the strength of the disciples' testimony has now also been verified.


Answering simplistic objections

Chapter 4:
Simplistic Objections

Is the prior too large, especially for a supernatural event?

One possible class of objections would try to argue that the prior probability for the resurrection wasn't small enough. So one may say:

"It's not just that people don't rise from the dead. NO supernatural claim of ANY KIND has EVER been validated in a controlled setting. Therefore the prior probability for the resurrection must be smaller than the value used in the calculation."

Well, let's again just give away everything the this objection asks for. So, take every human to have ever existed (1e11), and say that every single person has made 100 supernatural claims, all of which we have managed to test in "controlled settings" and have proven false. I will just ignore the fact that this level of testing simply hasn't been actually done. So, if I were to grant all that, the upper bound on the probability of the resurrection would drop to... 1e-13, which is 9 orders of magnitude LARGER than 1e-22, the value we actually used.

In fact, to demonstrate just how much we've already given away by setting the prior probability to 1e-22, consider the following scenario. Let's go ahead and say that every single mammal to have ever lived - estimated to be around 1e20 animals - have each made 100 supernatural claims, and that we have tested every single one of these claims and found them all to be false. Now, take a moment to actually imagine what this would entail: a time-travelling bunny would hop up to you and say "A mean old wolf tried to eat me, and I broke my leg while trying to get away - but then I was miraculously healed! And I was also blessed with this carrot!" And you'd respond, "Well, Mr. bunny, do you mind if I go ask Mrs. bunny, Mr. wolf, and your friends the sheep to see if they can verify your story? Because the other 99 times you told me something like this, it turned out to be false." So you would get into the time machine with Mr. bunny and his carrot to see if you can validate this supernatural claim.

It only is at this level of fantasy - with talking animals making supernatural claims, which you attempt to verify with your time-machine - that we finally reach enough of a sample size (1e20 mammals, 1e2 claims each) to reduce the prior probability to 1e-22. At this point, we're far into the realm of the absurd, and more than a dozen orders of magnitude past any semblance of empiricism. So the prior probability of 1e-22 we used is, as I already said, a far smaller value than anything any skeptic can rationally ask for.

Here is another objection along the "prior is too big" line.

"But science says that miracles can't happen; so whatever prior probability value you've set for the resurrection must have been too big to start with. If the conclusion to the calculation is that the resurrection actually happened, we must reduce the prior probability, so that we can arrive at a rational, scientific answer."

One wonders at how anyone can invoke "science" after abandoning empiricism and ignoring mathematical reasoning. This kind of statement betrays a willingness to pay lip service to math, reasoning, and science, while ignoring the conclusions that these fields actually lead to - all for the purpose of clinging to a bankrupt preconceived notion.

For instance, I have seen numerous skeptical arguments about miracles that mention Bayes' theorem and their prior probabilities. I have not seen a single one of these put an actual, numerical value to this prior probability. Among all the ones that I've seen, the argument has ALWAYS been "and since this number is going to be so small, it might as well be zero, although the value isn't actually, absolutely zero". So they claim to acknowledge that the prior probability can't be zero, while the argument functions as if it were zero in all circumstances. They thus pay lip service to probability theory, while ignoring it in practice, to reach their preconceived conclusions.

You must actually do the math. Use Bayes' rule. At the very least, don't just bring it up only to have your biases negate the whole point of using Bayesian reasoning. Ideally, try to assign actual values to the various probabilities and likelihoods, even if they're just order of magnitude estimates. Base these values on some kind of empirical data. And most importantly, don't just reject the conclusion because it didn't agree with your preconceived notions, or fiddle with the numbers to arrive at the conclusion you were looking for.

We'll continue our examination of possible objections next week.

Can human testimonies be trusted?

Could the disciples have been genuinely mistaken?

Another class of objections would try to argue that the Bayes' factors I used in my argument are too large. One possible objection along this line of thought might go like this:

"1e8 is a ridiculously large Bayes' factor for people's testimonies. People make mistakes all the time in their testimonies. Do you not know, for instance, how inaccurate eyewitness testimonies are? It is far more likely that the reports of Jesus's resurrections are mistakes of this type, rather than an accurate depiction of the events."

First, let's go over a few things before we tackle the specific issue on the reliability of eyewitnesses. The value for Bayes' factor that I used - 1e8 - is derived from the strength of a human testimony in general, with relatively few conditions attached to it. It is the typical value to be assigned for someone saying "yes, this really happened". Of course, if you start adding conditions to it, these will change the value of the Bayes' factor. So, I have no problem acknowledging that eyewitness testimonies can often be mistaken, and that it's in human nature to give flawed testimonies under certain conditions. In such conditions the Bayes' factor for a testimony must rightfully be severely discounted. However, one must also acknowledge that there are also conditions that dramatically enhance the value of human testimony - note the previous example of a recording a chess game, where a human testimony can have a Bayes' factor exceeding 1e120.

There is therefore bound to be a number of objections which effectively say "see how unreliable humans are (in these specific circumstances)!" What we must do, then, is to compare the circumstances in these objections to the actual circumstances surrounding the testimonies about the resurrection. We will see that, upon actually making this comparison, the testimonies for the resurrection are actually strengthened, rather than weakened, at nearly every turn by the specific circumstances surrounding them.

So, let's tackle the issue of eyewitness testimonies. The question of unreliable eyewitness testimonies typically come up in a courtroom setting, where a bystander is identifying someone they saw during an incident under investigation. A common example may have a policeman asking a witness, "now ma'am, can you point out which one of the fellows in that lineup was the one that pointed the gun at the cashier?"

Now, let's identify some of the common circumstances surrounding these events, about which such testimonies are made:
The witness is nearly always a bystander - a stranger who was previously not familiar with any of the actors in the crime.

The event in question often takes place in a matter of minutes, if not seconds. Witnesses are often caught by surprise - the crime takes place at its own pace, with no regard for making things easy for the witnesses. Indeed criminals often rely on the shock and the quick pace of the events to hinder possible identification and later prosecution.

There is often extreme stress placed upon the witnesses, who are fearing for their immediate personal safety. This may especially be the case if a weapon is present, which draws the focus of the victims or witnesses to it, and away from the proper identification of the perpetrator.

Related to the above, witnesses in such testimonies are often not primarily concerned with the identity of the perpetrator. In the moment, they are often simply shocked by the event, or mainly concerned about their bodily safety.
Now compare these to the testimonies about Jesus's resurrection:
Jesus was the most important person in the disciples' lives. He was explicitly more important to them than their family members or hometown friends. They had been around each other constantly for the last several years, and were familiar with one another as much as anyone can be.

Jesus's post-resurrection appearances occur multiple times, often in extended scenes where he converses with the disciples at length about what this all means. He eats with them, talks with them, and teaches them. Jesus furthermore specifically has these discussions for the benefit of the disciples, so that they can better understand his resurrection.

The pervasive mood during these post-resurrection appearances must have been awe and excitement. There is an optimal amount of stress for peak human performance, at a level which is neither too little (with accompanying boredom and lethargy) nor too much (with accompanying nervousness and panic). Speaking with the risen Christ must have put the disciples near this optimum peak, with an exhilarating atmosphere pervading every moment of their discussion.

The chief thought in the disciple's mind in each of these meetings must have been primarily about Jesus. 'Wow, it really is the Lord! He is risen from the dead! What could this all mean?' He commanded their wholehearted attention at each of these post-resurrection meetings.
So upon making this comparison, the result is clear. For each of the factors which causes courtroom eyewitness testimonies to be unreliable, the disciples' testimonies about Jesus are found to have the exact opposite property: they're testifying about someone they know very well (instead of a stranger), about events which happened repeatedly over an extended period of time (instead of being over in a flash), under the optimal amount of stimulation (rather than under crippling fear), with the person of Jesus as the chief object of their focus (rather than being shocked or focused on their immediate bodily safety). Insofar as the circumstances surrounding a typical courtroom eyewitness testimony cause them to unreliable, the same reasoning requires that the disciples' testimonies would then be especially reliable.

To put it simply, the example of unreliable courtroom witnesses only demonstrate how different the disciples' testimonies about the resurrection are. The disciples were not doing anything like saying "yes, that man with the red hair there is the man who pointed the gun at the cashier", with its accompanying uncertainty. No, their statement is rather more like a woman saying "yes, my husband really is the man I married at my wedding". Good luck finding many women who are mistaken about that.

Therefore, the Bayes' factor associated with the resurrection testimonies must be greater than they were in the unconditioned case. 1e8 may have seemed like an overestimate upon a superficial comparison, but a more careful consideration reveals that it is actually an underestimate: none of the factors that weaken a courtroom testimony are present, while all of their opposite qualities infuse the disciples' testimonies and correspondingly strengthen them.

We will continue with more objections next week.

Or actively deceptive?

Yet another class of objections may argue for 1e8 being too large, on the basis of people being intentionally deceptive rather than being mistaken. It may go like this:

"1e8 is a ridiculously large Bayes' factor for people's testimonies. People lie all the time. Do you really think that only 1 out of 1e8 things that people say are lies? There are conspiracies, con artists, and fame seekers everywhere, at all times. What makes you think that the disciples reporting on the resurrection were not just one of these people?"

The objection here, and its answer, is much the same as before. Yes, people lie, or are otherwise unreliable, in some circumstances. These circumstances rightly require us to adjust the Bayes' factor downwards. But the comparison of such circumstances with with what the disciples actually faced will only reveal their vast differences. If you think that people are likely to lie under certain circumstances, you must then therefore think that the disciples were highly likely to be truthful about the resurrection, due to the absence of these circumstances.

So, taking lottery winners again as an example: if someone claims to have won the lottery, their claim should be given about a 1e8 Bayes' factor. But what if they then go on to say that they've left their winning ticket with a Nigerian prince, and that they would share their winnings with you if you would only give them $5000 to cover their travel expenses to retrieve the ticket? Well, now the Bayes' factor drops precipitously, down towards zero.

However, what if the supposed lottery winner instead gives lavish gifts to their friends and family, buys a new house, then hires a financial adviser to discuss the tax implications of their sudden windfall? Then the Bayes' factor would dramatically increase, towards values like 1e120.

So then, what are the circumstances under which people are likely to lie? And in contrast, what are the circumstances that the disciples faced?

Well, people often lie for material gain, as in the above example of a con artist. The disciples, however, did not accrue wealth by claiming that Jesus had risen from the dead; in fact the very nature of their claim made this outcome highly unlikely, with the emphasis on serving the poor and a general disdain for worldly gain. If money was their goal, this was certainly the wrong way to go about it.

People may also lie under social, psychological, or physical pressure, as in the cases of false confessions obtained under harsh interrogation or torture. The disciples, however, resisted such pressure, and held on to their testimony under immense opposition of all kinds. The imminent possibility of persecution is a constant theme throughout the entire New Testament. In fact, many of the early Christian leaders underwent torture and martyrdom, including all three of the named witnesses I used in my calculation (James, Peter, Paul). We know how effective such treatment can be in eliciting false confessions even from their modern victims. We must therefore consider anyone who resisted the far harsher ancient versions of these treatments to be exceptionally trustworthy.

One may argue that at least the negative social pressure from society at large may be made up for by the approval from the close-knit Christian community. But this simply does not apply. Again, among the three named witnesses I used in my calculation, only one (Peter) was originally one of Jesus's disciples. James and the rest of Jesus's family are considered to have been in a somewhat disharmonious relationship with Jesus before the resurrection, and Paul was a complete outsider - an early persecutor of the church, whose personal and social identity was very much set in opposition to Christianity. So in a majority of these cases, the social pressure would have gone the other way: they would have ample reasons to reject the resurrection. Their testimonies in spite of this, therefore, must be counted as being much more reliable than the average.

Incidentally, if you thought that I forgot to adjust my calculations for the fact that the testimonies are not independent, this is why - the three named witnesses in my argument ARE largely independent; they come from very different backgrounds and met the risen Christ under different circumstances. Especially in Paul's case, if anything you'd expect his testimony to be anti-correlated with Peter's. For the other witnesses where dependency is expected, I explicitly called it out and severely discounted the Bayes' factor values in the calculation.

Now, back to the subject of lying: people may also lie for fame - they claim to have achieved something remarkable or to be someone special. But as we have just seen, the fame that came with proclaiming the resurrection would have been exactly the wrong kind of fame; the witnesses would have been shunned both by the Roman and Jewish society at large, and in many cases by their immediate social circle. Furthermore, it is the nature of fame to be fleeting; few would continue to lie for fame, in the face of intense opposition, for decades at a time, long after the shock of the initial claim wore off, to the point of death. Indeed, if the witnesses were fame-seekers of this type they would have done quite well by recanting the resurrection at the last minute and becoming a kind of whistle-blower for this deception that Christians pulled over the world. And yet, the witnesses did no such thing; they all died as martyrs.

People also sometimes lie for a cause. If they believe that some agenda is good and important, that may cause them to be deceptive "for the greater good", to advance that agenda. But this is impossible given the theology of the early church. Jesus was the greatest good; his resurrection was the most important event in the whole world. There was nothing greater which would be worth lying about the resurrection.

In all this, the actions of the witnesses were in perfect accord with their genuine belief in the resurrection. They had no reason to lie and every reason to tell the truth. We, also, have no reason to believe they were liars and every reason to believe that they were truthful.

So, it is true that men often lie. But this is a shallow observation. Upon considering the actual, specific circumstances surrounding the resurrection testimonies, we find that they are diametrically opposed to the circumstances conducive to lying. Therefore, the observation that "men often lie" only serves to enhance the trustworthiness of the witnesses to the resurrection, by pointing out how different these witnesses are from typical liars.

We ought to have reduced the Bayes' factor for the resurrection testimonies down from 1e8 had we found the surrounding circumstances conducive to lying. But since the opposite has happened, we must therefore increase the Bayes' factor. 1e8 is a drastic underestimate of its true value.

We will continue with more possible objections next week.

Or actually crazy?

Another class of objections would just argue that the witnesses to the resurrection were crazy:

"Obviously anyone who claims that they saw someone coming back from the dead is crazy. How can we take their stories about these outlandish miracles seriously? Clearly there was something mentally wrong with these people, and we ought to dismiss their 'testimonies' as the ramblings of the insane or the schizophrenic."

By now, it ought to be obvious that I'm going to handle this objection like all the others. Did the witnesses to the resurrection act like they were crazy? Did they exhibit the typical behaviors of the insane or the schizophrenic? If they did, we should rightly lower the Bayes' factor for their testimonies from the relatively unconditioned value of 1e8. But if they did not, then by the same logic we must increase the Bayes' factor.

This investigation is straightforward enough: read the New Testament, and look for symptoms of mental illness in areas that are not directly related to supernatural claims (one must be careful about circular reasoning). So, does the New Testament read like the work of a schizophrenic? Does it seem to describe people who were afflicted by mental illness? Would you say, for instance, that Peter's sermon at Pentecost exhibits problems with attention or memory, or that Paul's letter to the Romans demonstrate disorganized thinking?

In fact, apart from the supernatural components, I have not heard of anyone citing any part of the disciple's work in the New Testament as being characteristic of mental illness. If there is such a passage, I'd love to know about it. Can anyone point to a verse and say, "here is where Paul shows clear signs of psychosis", or "this is where Peter displays the classic symptoms of schizophrenia"? It says a great deal about the "insanity" accusation that the only evidence they can find for it are the very parts that make up the question at hand, the very parts they object to. In short, the objection effectively only amounts to saying "I disagree with these people on these points, so they must be crazy!"

On the other hand, there are plenty of reasons to think that the witnesses to the resurrection were of sound mind. Remember, they were the organizers and leaders in the early Christian church - a movement that spanned their known world. Furthermore, recall that they were successful beyond any naturally possible expectations: Christianity has lasted thousands of years until the present day, multiplied wildly, and now spans the whole globe. Can anyone give any example of an organization run by insane people that was even a millionth as successful?

In particular, the ideas behind this organization - that is, the theology of the early Church - are readily available to us as the text of the New Testament. They are the most read, discussed, studied, and applied texts to have ever been written. If you're reading this blog you're also free to go and read the New Testament. Does it seem like the work of the insane? What work by any mentally ill persons has ever reached a fraction of its stature?

So the conclusion is clear enough. Once again, upon actually considering the facts surrounding the resurrection witnesses, we find that they do not correspond at all to the scenario in the objection. The disciples display no sign of insanity, instead demonstrating many characteristics of sound and acute minds. So, according to the very logic embedded in the objection itself, this must again increase the Bayes' factor of their testimonies.

We will continue with more next week.

Or some combination of the above, or something else entirely?

Here is another typical attempt to deny Christ's resurrection:
"It may be that some of the disciples were crazy or especially grief-stricken after Jesus's crucifixion. This lead them to see some vivid visions of Jesus, which they related to the other disciples. Some of these other disciples, who had not seen the visions themselves, then spread the story about the 'resurrection' based on the vision of these few crazy people. Then, a few other disciples, who were dissatisfied with Judaism, formed an opportunistic conspiracy to start a new religion based on these budding stories about this 'resurrection', and that's how Christianity started.

Or, it could have gone another way. A few disciples wanted to start a new religion and formed a conspiracy. After Jesus's death, they suggested to some of the other, more gullible and mentally unstable people that Jesus rose from the dead. With a little faking of evidence, social pressure, and the power of suggestion, they eventually got enough of the other disciples to say that they saw the resurrected Jesus themselves. From there, the resurrection became part of their faith narrative, and that's how Christianity might have started.

There are dozens of other possibilities like these - it doesn't have to be that everyone was lying or crazy. We just need the right combination of lies, mistakes, and insanity at the right times and situations for Christianity to start. Surely, it is more likely that one of the many possibilities represented here lead to the belief in the resurrection rather than for Jesus to have really come back from the dead."
This is merely an attempt to muddy the waters by complicating the issue.

First, note how weak this argument is, even if we were to grant it everything that it asked for. Remember, the odds for the resurrection are currently at 1e32, so the odds against it are therefore at 1e-32. Now, we'll allow for each independent objection to count as having the full weight of these odds. Never mind that many of these objections contradict one another and therefore reduce the probabilities of the other objections (increasing the probability for 'insanity' decreases the probability for 'conspiracy', because a conspiracy is less likely to succeed with insane people in it). We'll just ignore that. Never mind also that these complex speculations are naturally less likely because of their complexity. We'll also ignore that as well. So, if we can think of a hundred such objections, each of which carries the full weight of the 1e-32 odds for 'no resurrection', the final odds for the resurrection would drop all the way down to... 1e30.

In short, if you're serious about this approach, go ahead and write out a billion independent objections of the kind demonstrated above. That would drop the odds for the resurrection to 1e23, and it might then merit a footnote as something that someone might want to look into sometime.

Conclusion: there are no reasons to consider any of these. And even so...


Chapter 5:
The strength of a Bayesian argument: why none of these objections work.

Everything is already taken into account

After hearing many objections in succession as we just have, it's easy to lose sight of the big picture. For instance, one may fall into the trap of thinking that if even one of these objections has even the slightest chance of being true, the argument would fall apart. But is that really the case? If the disciples had even the slightest chance of being crazy or mistaken or deceptive about the resurrection, would that cause the whole chain of reasoning to break and the case for the resurrection to collapse?

This is where it's useful to remember the big picture. You see, a standard deductive argument does work like that - A and B together lead to C, which lead to D, which then leads to the conclusion. For such an argument, all of its premises must be entirely true and each step of its reasoning must be completely correct. Anything else invalidates the whole argument. That is why a barrage of objections can sometimes succeed against such an argument, or at least cast doubt on its soundness.

But my argument for the resurrection is not a deductive argument. It is an order-of-magnitude probability estimation argument using Bayesian reasoning. The objections against it can take two forms: you must either claim that I'm misusing the mathematical apparatus (that is, Bayes' rule), or disagree with my estimated values for the probabilities.

If you think that I've made a mistake in applying Bayes' rule, then by all means point it out. Otherwise, the objections against it all come down to wrangling over the probability values. The point here is that in such wrangling need not produce absolute certainty. The argument does not depend on it. I do not need to claim, for example, that there is absolutely no chance that the disciples were lying. Having demonstrated that the Bayes' factor for a typical, relatively unconditioned human testimony is around 1e8, I only need to demonstrate that the disciples are not more likely to be liars than such a "typical" person. In fact, anything which suggests that the disciples' honesty exceeded that of the "typical" person actually strengthens the argument beyond its original form, by increasing the Bayes' factor. This is what has actually happened upon the examination of every single objection thus far.

Furthermore, even if one of the objections were to "succeed", it would not be a fatal blow to the argument; we would merely have to re-calculate the final odds. So, for instance, let's say that the objection about the disciples being mistaken somehow "succeeds", and it results in the Bayes' factors from their testimonies dropping from 1e8 to 1e4. In fact, let's say that these objections are so wildly successful that we must take the square root of every single Bayes' factor we used. Such an instance would not cause the whole argument to be simply invalid. Instead, we would just have to recalculate the final odds with the new numbers. We'd find that the final odds dropped from their original value of 1e32 to 1e5 - corresponding to around a 99.999% chance FOR the resurrection still having occurred.

So, let's be clear about the effect of these repeated objections. Our argument is not a deductive argument. If it were, any objection might cause the whole argument to be invalidated if we can't demonstrate with absolute certainty that the objection is false. It is easy to think this way when you hear many objections in succession - that one of them must eventually get through a chink in the armor and deliver the fatal blow. But it simply does not apply to our case.

Rather, our argument is an order-of-magnitude probability estimation using Bayesian reasoning. From the beginning, it is strong enough to survive multiple successful objections against it. We are then fielding objections against this argument. If an objection succeeds, that would decrease the power of the argument - but if it fails it must correspondingly increase the power, by demonstrating that the resurrection witnesses were more honest, correct, and sane than what we had initially assumed. And this latter case is what has happened in every objection thus far.

So, we've started from an incredibly strong original argument, and each of the repeated objections have only strengthened it further. We will continue to look at more opportunities for strengthening next week.

Only evidence moves the odds. Speculations do nothing

But more importantly, this kind of objection is simply, fundamentally wrong: it would not fly in any other investigation into a personal testimony, because it completely ignores the rules about how we evaluate evidence in a Bayesian framework.

Imagine, for instance, that your friend claims to have been struck by lightning. You've taken stock of this claim and have decided to assign it a Bayes' factor of 1e8. But then you say, "well, you may be just a little crazy. And you might have had a nightmare about a thunderstorm last night. Then you might have gone to a hypnotist and who had you recall your dream, which you're now confusing with reality. Or maybe it was the hypnotist who planted the suggestion in your mind first and that caused your nightmare. Really, it might have been any of these things - and isn't it more likely that at least one of these possibilities is true, rather than for you to have been actually struck by lightning?"

Should you or your friend then discount the previously assigned Bayes' factor in light of these new possibilities? Absolutely not. The thing to note here is that the Bayes' factor ALREADY includes all of the ways that this claim may be wrong. It is the numerical estimation of the weight of evidence for a human testimony, and as such already inherently includes the possibility that the evidence may be misleading.

Having established its value, it is simply incorrect to further modify it with no evidence, based on enumerating possibilities that were already included in its evaluation. Your friend's proper reply to your wild speculation would be to say, "what makes you think that I had visited a hypnotist or had a nightmare? Of course, anyone might be wrong about anything in any number of ways - but don't you already know how much you trust me? How does a list of ways that I might be wrong, with no evidence behind any of it, make you trust me less?"

Again, this goes back to the "barrage of objections" tactic I mentioned earlier, and why it fails against the resurrection argument presented here. The trap is to get you to think of the many possible objections against the argument, then confuse you into thinking "at least one of them must work!" It requires you to forget that the argument here is fundamentally immune to that line of attack. Once again, my argument is not a deductive argument. It is an order-of-magnitude probability estimation using Bayesian reasoning. It starts with a prior probability, and then it modifies that probability based on the evidence. So, is there any evidence that your friend had a nightmare about a thunderstorm? Then it is proper to include that evidence to re-calculate the probability of the lightning strike. Is there any evidence that a crazy group of disciples reported on the resurrection, which then got hijacked by a conspiratorial group of disciples? Then it would be proper to include that evidence to re-calculate the probability of Christ's resurrection. However, in the absence of such evidence, the mere existence of that possibility should not change our calculations. Such possibilities are already included in the initial calculation.

Let me give an even simpler example. Suppose you flip a coin, then cover it up so that you don't know the outcome. Not having investigated the coin all that carefully, you assume that the probability of it turning up 'heads' is 0.5. Now, someone comes up to you and says, "but consider all the ways that it may turn out to be tails. It might have hit the tabletop, flipped three times after the bounce, landed on its edge, then fallen over to show tails. Or it may have flipped fifteen times before the first bounce then landed flat with the tails side up. In fact, if the coin's leading edge strikes the table at 15 degrees with an angular velocity of 12 rev/s and a downward linear velocity of 2 m/s, it's guaranteed to end up tails. And this is only a small sample of the innumerable ways for you to get tails. Given all these different ways, shouldn't you decrease your 'heads' probability?"

The answer, of course, is that you should not. You should only consider the evidence that you DO have in modifying your probability. You must leave alone any evidence that you MIGHT have. So, in the absence of any evidence, the probability for 'heads' is still 0.5, and the innumerable ways that the coin might turn out to be 'tails' does nothing to change it. Now, it may be that you recorded the first part of the coin flip in slow motion, and it turns out that the coin did indeed strike the table at an angle of 15 degrees for its leading edge, with an angular velocity of 12 rev/s and a downward linear velocity of 2 m/s. That would be evidence. That would cause the probabilities to change. But the mere possibility of this happening, in the absence of the actual evidence, does not change the probability.

Here is the evidence that we DO have: numerous witnesses gave their earnest, personal testimonies, saying that they personally saw the risen Christ. We know how to numerically evaluate such evidence. Earlier in this argument, we have already numerically taken into account the many ways that they may have been wrong, whether through honest mistakes, deception, or insanity. We have no evidence that anything like the speculations of the skeptics have taken place, and the mere possibilities for these speculations cannot change the probabilities. The odds of the resurrection, even with the most conservative estimates, remains 1e32, at a minimum.

We will continue with other parts of the argument next week.

The lack of evidence against the resurrection

We've just touched on the lack of evidence for doubting the resurrection. This is important, because it allows me to answer all the other skeptical arguments and distinguish my argument from them. A skeptical reader may wonder whether I've ignored any evidence against the resurrection, or how I would answer this or that argument from this or that website. A significant part of my reply would be that there is no evidence against the resurrection.

Let me reiterate and clarify that, because it's important. There is an utter lack of evidence for disbelieving the resurrection: literally every single record we have from the people who were actually connected to the event to any reasonable degree ALL portray the resurrection as something that actually happened.

If you believe in the resurrection, you have the unanimous support of all the people who were actually close to the event and would know for certain. If you disbelieve the resurrection, literally every piece of evidence - every single testimony of every single person who ever testified about the actual event - is against you.

So, I'm not being selective about the evidence. There is nothing to be selective about, because there is literally no evidence for the opposing argument. This is why I'm fundamentally unconcerned about the arguments against the resurrection: because they have no evidence. The only thing I've done in choosing my evidence was to handicap my own argument, by only using a tiny fraction of the total evidence available.

If there were any evidence against the resurrection, I'd be glad to incorporate it into the calculation. I've already said elsewhere that a sufficiently strong evidence against the resurrection can falsify the whole hypothesis for me - if, you know, such things actually existed.

So, does anyone know of a cave in Israel that houses Jesus's mummified corpse? By all means, tell me about it. Is there an ancient manuscript that exposes the disciples' conspiracy to fake the resurrection? Let me know. Is there a record of a Roman interrogation where an apostle confesses to having made up the whole resurrection thing? Is there an epistle where a disgruntled disciple warns the others about staking the faith on a schizophrenic woman and her crazy resurrection story? Is there any record of a psychoactive plant in first century Jerusalem that causes vivid mass hallucinations about the recently dead? Is there a complaint from Jesus's family about how his message has been hijacked by a bunch of lunatics and their crazy resurrection story?

You see, nothing remotely like any of the above actually exists. There is literally zero evidence for disbelieving the resurrection.

This is why every single skeptical attempt at explaining the resurrection relies entirely on ignoring the existing evidence, and making stuff up instead. They have no other options, because they have no evidence on their side. That's why the only thing they can do is to ignore the existing evidence, and make stuff up.

So, when they say that Jesus's resurrection was a myth that grew over time to be accepted as fact, they're ignoring the existing evidence that says that the resurrection was at the very core of Christianity from its inception, and making stuff up instead about how a myth might have eventually gained enough traction to be accepted as dogma.

When they say that Paul might have converted because he already had second thoughts about Judaism before encountering Jesus on the road to Damascus, they are ignoring the existing evidence in Paul's own testimony, and making stuff up instead about what they think went on inside Paul's head.

When they say that the early Christians didn't believe in a real, physical resurrection, they are ignoring the existing evidence that unanimously say that Jesus's body was missing from the tomb, and are instead making stuff up about what they think the early Christians really thought.

When they say that Jesus might not have really died, but only swooned, they're ignoring the existing evidence that clearly presents Jesus's death, and making stuff up instead about the combination of circumstances that might have allowed Jesus to survive a crucifixion.

When they say that the post-resurrection appearances were only visions or hallucinations, they're ignoring the existing evidence that unequivocally states the physical nature of Jesus's new body, and making stuff up instead about the disciples' mental conditions.

When they say that the gospel writers were only interested in the theological and literary dimensions of their story, and showed no concern for the truth, they're ignoring the existing evidence from these writers themselves that directly contradicts them, and making stuff up about the writer's "true" motivations instead.

So, let's not be distracted by such made-up speculations, and instead stick to the existing evidence that we do have. Remember the outline of the argument at hand. We are using Bayesian reasoning. We start with a prior probability for the resurrection, and modify it according to the existing evidence that we actually have. There is no place in this calculation for speculations about what evidence we might have if some made-up stuff happened instead. Such speculations cannot modify the probability, for any possibility for such made-up scenarios are already included in the calculation: the inherently unlikely nature of the resurrection is already included in the prior, and the possibilities for the disciples being wrong are included in the Bayes' factors. Upon carrying out this calculation, using absurdly conservative values, we find that the odds for the resurrection are at 1e32 to 1, at a minimum. Therefore, Jesus almost certainly rose from the dead.

There is more to come in the coming weeks - starting with the next post.

(part 11, addressing "disciples stole the body" rumor)

Conclusion: The Bayesian argument takes it all into account



Addressing all possible alternatives

Chapter 6:
Time to address the crackpot theories

Why we now turn to crackpot theories

We can now be very confident that Jesus rose from the dead. Our previous calculation which first gave us this confidence has now been verified in multiple ways, using completely different methodologies - by double-checking with the historical background of non-Christian resurrection stories. Everything checks out, and all the numbers are in harmony.

But... all this has been computed under the assumption that there isn't any extreme dependence in the disciple's testimonies. We've already accounted for "normal" dependence, like ordinary social pressure or group conformity. But we have not yet accounted for the possibility that the entire set of testimony about Jesus's resurrection might have been been engineered to be in agreement by some unknown force. That is to say, we've been discounting crackpot theories - like a conspiracy by the disciples to steal Jesus's body, or an alien mind-controlling all the witnesses to the resurrection.

Ignoring such theories is fine and good, as long as both sides of the debate are agreed in dismissing them. Most doubters of the resurrection do not subscribe to these extreme theories, so carrying out our calculations in this way up to this point was still productive. However, they're now facing a double-checked Bayes' factor exceeding 1e54 for the resurrection. This makes the posterior probability against the resurrection so tiny, that the small prior probability assigned to crackpot theories now seem much larger in comparison. Someone set on disbelief can no longer ignore these theories. Indeed they have no other choice: they must fully embrace these crackpot theories.

We will begin to address such theories starting next week.

(addressing "disciples stole the body" like theories)
(part 11, 30)

Examining crackpot theories, in general

Let us examine this general class of theories, that postulate a near-total interdependence in the evidence against them. What kind of theories are they? What are their properties? Is it fair to characterize them as "crackpot" theories?

Now, note that such theories requires a conspiracy of some kind, almost by definition. Near-total interdependence means that what appeared to be many pieces of evidence was really just controlled by a singular false entity, which manufactured all the other pieces of evidence. Whether this source was a group of disciples or an elite Roman secret society or some space aliens or whatnot doesn't particularly matter - All such theories share the following traits.

The first thing to note about such theories is that they have very low priors probabilities to begin with. Indeed, among those skeptical of Christ's resurrection, a theory of this type is almost never their first choice. Few people want to be labeled a conspiracy theorist, after all. The skeptics want the resurrection testimonies to have been produced "naturally". They'll invoke known social phenomena such as myth generation over a long time, or religious fervor or delusion. They want such ordinary explanations to be a plausible way to generate the resurrection testimonies. Of course, what we've demonstrated thus far is that such explanations are in fact not plausible - that they're faced against a Bayes' factor of more than 1e54.

Maybe some people will say that they'd rather be a conspiracy theorist than believe in the resurrection. But even so, such people only say this as a backup, while still trying to argue for a more ordinary explanation.

So, conspiracy theories and other similar hypothesis have low prior probabilities, even in the mind of skeptics. This is appropriate, as conspiracies are in fact very rare.

Secondly, these 'near-total interdependence of evidence' theories are designed to ignore the evidence. They are chosen precisely because they allow their adherents to say "but that's exactly what they want you to think!" to any evidence you bring against them. It's important to note that this is not an accidental, fortuitous property of these theories. 'Near-total interdependence of evidence' is the defining feature of such theories, and it's precisely that feature which allows them to dismiss all the evidence which would weigh against more likely theories.

In combination, the above two facts mean that such theories cannot really hope to win the day. Since they start with a low prior, and are designed for ignoring the evidence, they cannot really hope to prevail - they need evidence to increase that low prior probability, but they're designed mostly to ignore evidence.

Note that, when a conspiracy theorist ignores evidence by saying "that's exactly what they want you to think!", this doesn't actually help the theory. It merely turns a piece of evidence against the theory into no evidence. Yes, the conspiracy theory has "explained" the evidence, but only about as well as the rival theory. The Bayes' factor therefore stays around 1, meaning nothing has changed on that front, and the probability for the conspiracy theory remains at its low prior value.

But, such evidence does still hurt they conspiracy theory, because the prior probability itself is now a lower value. A greater conspiracy that explains more - one that is more vast and has planted more evidence and covered it up better - is a priori less likely to have come about than a lesser conspiracy. So a piece of evidence that the conspiracy has to dismiss does still hurt the theory. The hope of the conspiracy theorist is that this harm in the prior probability will be less than the exponential rate of harm that a fully independent piece of evidence would normally cause.

So the most such a theory can realistically hope for is a kind of non-total loss, where they lose less quickly and hope to say "at least it's not impossible!" at the end.

Now, there are very particular kinds of evidence that does help them - the ones that specifically demonstrates a conspiracy. Something like a document from a secret meeting that lays out the nefarious master plan would work. But, of course, for a vast majority of these theories, such evidence does not exist.

So, given all these traits - given that they are highly unlikely theories that are designed to ignore the evidence, with little chance at any positive evidence for them - I think it's fair to call them crackpot theories.

In fact, in the specific case of Christ's resurrection, the situation is even worse for these theories - for there are many factors within the resurrection testimonies that are highly effective in working against them. We will examine these in the next post.

Conclusion: we now turn to examining even such theories


Chapter 7:
The "skeptic's distribution" approach

Use the historical data to construct the skeptic's distribution

Can we quantitatively tackle things like conspiracy theories? What do we do about the interdependency of evidence? One can already imagine the objections to any such attempt. Every assumption would be questioned, and every ridiculous possibility brought up demanding a full numerical treatment. Even if a traditional conspiracy were to be fully debunked in a numerical argument, a skeptic would just weasel the argument to be about a "groupthink induced by religious fervor" instead, and when that got debunked, they would just move on to "have you considered aliens?" Indeed, such weaseling is often the point of bringing up things like conspiracy theories in the first place: not to actually advocate for them, but to make the calculation appear intractable.

But I did say in my last post that I will approach this problem quantitatively - and that's exactly what I'm going to do. Furthermore, my argument will take EVERYTHING into account - government conspiracies, religious groupthink, practical jokes by aliens, everything. Every single possibility for every conceivable degree of evidence dependence will be fully considered.

In addition, empirical evidence will be the foundation of my whole argument. That is, in fact, the key that makes it totally comprehensive. Do you remember the following graph?

That is the level of empirical evidence that history has actually recorded for the resurrection of various individuals. It's a partial histogram - note the differing number of people with different amounts of evidence for their resurrection. This suggests a probability distribution.

Of course, the graph above isn't the complete record of everyone - it's a small sampling of some people who have the most evidence for their resurrection. But if we had a complete record, we could get a very accurate model for their underlying probability distribution. What would that probability distribution represent?

If we exclude Jesus and the other Christian resurrection reports, the probability distribution we get would be the EXACT model that an empirical skeptic of Christianity MUST use, in predicting the likelihood of a resurrection report. Essentially, the idea is that we can calculate the probability of getting a certain level of evidence for a resurrection, based on how frequently similar reports have come up in history.

Note that, because the raw data is gathered from empirical reports collected in history, this automatically takes things like conspiracy theories into account. The possible interdependency of the evidence is fully included in this model. So you think that a great deal of evidence can be built up through a conspiracy, because the evidence doesn't have to be independent? The distribution includes all such evidence-manufacturing conspiracies that actually existed in history. You want to switch your argument to a religious mass delusion instead? The result of all such mass delusions are also included, at the level of evidence that they actually generated in history.

How about something that has probably never happened in history at all, like an alien resurrecting someone as a joke? Even these possibilities are included, through at least two mechanisms. For one, there are a great many multitude of such unlikely scenarios - and at least one of them might have actually occurred in history, even if a specific one of them was unlikely. So they would have recorded their evidence in aggregate. And secondly, even if such unlikely scenarios never occurred, they can still be accounted for in the modeling of the probability distribution from the samples we actually have. As an analogy, if you were to model people's heights by sampling a thousand people, you will still deduce that human heights follow a roughly normal distribution, and can thereby figure out that there may be someone out there who's at least 7 feet tall, even if such a person was not in your sample.

So you see, this method does in fact take everything into account. It does generate the exact model that an empirical skeptic of Christianity must use. That's the great thing about arguing from empirical, historical records. You can bypass all the difficult and controversial calculations about the probabilities of conspiracies, or the precise degree of dependence among the evidence. All of that automatically gets incorporated into the historical data at their actually correct historical values, and all we have to do is to read off the final result.

Once we have this "skeptic's distribution", the rest of the calculation is fairly straightforward. We can calculate the probability of generating a Jesus-level of resurrection evidence, from the point of view of the skeptic's hypothesis. From a Christian perspective, this probability is within an order of unity, so the skeptic's probability then essentially becomes our Bayes' factor. We then simply see if this Bayes' factor is enough to overcome the low prior probability for a resurrection.

Now, I will use 1e-11 as the prior probability for a resurrection. This is higher than the 1e-22 that I had used earlier - but recall that I only got 1e-22 by simply giving away an additional factor of 1e11 for no reason. I can't afford to just give that away when faced against something as insidious as conspiracy theories. 1e-11 is still smaller than anything that a skeptic can demand based on empiricism, and the rest of the argument will be constructed so that even the minimum Bayes' factor will exceed 1e11. The actual Bayes' factor may exceed 1e22 by the end - I just can't be as precise in my calculations at that level.

We will start constructing this "skeptic's distribution" in the next post.

Properties of the skeptic's distribution: power law.

Recall that we're constructing a "skeptic's distribution" - the probability distribution of generating a resurrection report with a certain level of evidence. We will construct it from historical, empirical data. This allows us to bypass the mess of trying to compute everything from first principles, and ensures that the this is the correct distribution - a skeptic cannot reject it without rejecting history or empiricism.

So, what form should this "skeptic's distribution" take?

How about using a normal distribution? Well, that would be plainly ridiculous. The data we have thus far has Jesus (very conservatively) having 24 times more evidence for his resurrection than anyone else in history. Our goal is to get the probability for something like this happening.

But if we used a normal distribution for the "skeptic's distribution", this could essentially never happen. Recall that human heights roughly follow a normal distribution. Then, our problem would be analogous to looking for someone 24 times taller than anyone else in history - that is, someone well over 200 feet tall. The probability for something like that is essentially zero. So if we chose the normal distribution, we'd essentially be dooming the skeptic's case from the start.

The same is true for an exponential distribution. An exponential distribution decreases in its probability value by a constant factor for each unit of increase in its domain. As the domain is "level of evidence" in our problem, this means that each piece of evidence would multiply the probability values. That is to say, we'd be treating each piece of evidence independently. And we already saw that even with a reasonable degree of dependence factored in, the probability values already reached numbers like 1e-54, again dooming the skeptic's case.

This is a testament to how quickly these distributions decay as they extend to the right. Their right tails are so "stubby" that the maximum values of their samples are strongly restricted, and getting something 24 times greater than that maximum is essentially impossible. Picking any such distribution would not be taking into account the dependence of the evidence, and would unfairly doom the skeptic's case from the start.

Rather, we need a distribution with a "long tail" - something that has a chance for a new high record to beat the previous record by factors like 24. Something that decays slowly enough that its probability values remain non-negligible as we move further to the right. The distribution should still be realistic and have some justification for being selected, but we want to give the skeptic the best chance.

Taking all that into account, I have chosen a power law function for the "skeptic's distribution". This should not be a surprise - indeed anyone familiar with the statistics of human behavior might have guessed it from just the histogram we're trying to fit:

What makes a power law particularly appropriate? Well, for one, power laws are the quintessential long-tailed distribution. They have one of the longest possible tails, and are fully "capable of black swan behavior", according to Wikipedia. They can easily have tails so long that the overall distribution has an undefined (that is, infinite) mean. In fact, power laws, as mathematical functions, can decay so slowly that it's not allowed to be a probability density function, because the area under their curve can diverge. One can hardly ask for a more slowly decaying function than that. So this gives the skeptic the best chance at naturally generating a Jesus-level of resurrection evidence.

There exists distributions that decay even more slowly than a power law, but they're rare, obscure, and have no relation to what we're doing. By contrast, power law distributions are ubiquitous in human behavior. They form the basis for the well-known Pareto principle, and they capture the "dependency of evidence" factor we're currently trying to model.

For example, the distribution of income among people follows a power law. A few people, out at the long tail, have a great deal of wealth, because rich get richer - that is, because how rich you get depends on how rich you already are.

The size of cities also follows a power law. There are a few very large cities out at the long tail, because your chances of moving to a city depends on the number of people who already live there.

The number of links to a website follows a power law. There are a few, very popular websites out at the long tail, which have a lot of links to them. This is because a site's chances of getting a link depends on its popularity - that is, on the number of links it already has.

Don't let the specificity of these examples fool you. There are many, many more. Power law distributions are, as I said, ubiquitous in human behavior. They will frequently come up when one human behavior depends on the same kind of behavior, either by others or by the same person.

So it is entirely appropriate that we use a power law to model the level of evidence for a resurrection report. There will be relatively few reports out at the long tail, like the "resurrection" of Apollonius or Krishna. In the context of things like conspiracy theories, this is because the chances of generating an additional piece of evidence depends on how much evidence it already has.

So there are excellent external reasons and examples to expect that the "skeptic's distribution" will follow a power law. Furthermore, power laws give one of the best possible chances for the skeptic's case, having a very "long tail" and allowing for a "black swan" event like the level of evidence in Jesus's resurrection event.

We will proceed to nail down the specifics of this power law distribution in the next post.

Details of the distribution: generalized Pareto distribution and its parameters

We've decided on a power law as the general form of the "skeptic's distribution".

The details of the distribution near zero will not particularly matter. We're more concerned about how rapidly it decays at very large values. This allows us quite a bit of leeway in choosing the specific form of the power law distribution, as they all decay similarly as we move along the tail off to the right.

For this reason, I've chosen the generalized Pareto distribution as the specific form of the "skeptic's distribution", guided chiefly by the straightforward interpretation of its parameters. But the choice here will not affect any conclusions. Any other power law distribution would give the same results.

The generalized Pareto distribution is characterized by three parameters: location, scale, and shape. The location parameter determines where the distribution starts. It's where the probability density of the distribution is the largest. As the vast majority of humans have zero evidence suggesting that they rose from the dead, the location parameter should obviously be set at zero.

The scale parameter is irrelevant; it only controls how far the distribution should multiplicatively scale in the horizontal direction, and can be arbitrary changed by changing the unit of evidence we use. As we'll consider all the evidence for our the resurrection reports relative to one another (for example, as a fraction of the amount of evidence for Christ's resurrection), the value for the amount of evidence in some specific units never enters the picture. So we'll just set this parameter to 1, or to whatever is convenient for visualization, and forget about it.

The shape parameter is the interesting one. It's what we really care about. It effectively determines the power in the power law, and controls how quickly the function decays as the amount of evidence increases.

For example, this is what the tail end of the distribution looks like with various shape parameters:

In each case, the distribution has been scaled so that the total probability to the right of the grey line (at x=1) is 1e-9. Essentially, x = 1 is where you would expect the maximum value out of 1e9 samples to appear, corresponding to the level of evidence for the resurrection of a figure like Apollonius or Aristeas.

Note the different rates decay. With the shape parameter at 0.2, the probability density drops to practically zero as we move to larger x values. There is essentially nothing left by the time we've moved to x = 24, even if we integrate out to infinity. Therefore, if this were the final form of the "skeptic's distribution", the probability of generating a Jesus-level of evidence for a resurrection would be essentially zero.

However, with the shape parameter at 2, we see that the decay rate is much slower, and there is a good amount of probability even out at x = 24 and beyond. If this were the "skeptic's distribution", it would have a good chance of generating a Jesus-level of evidence for a resurrection, even if that level were 24 times higher than the runner-up.

A shape parameter of 20 decays more slowly still. It's hardly decaying at all by the time it reaches x = 24. In fact, it decays so slowly that the blue curve with the shape parameter of 2 will eventually move below it. If this were the "skeptic's distribution", it would have a non-negligible chance of generating an event at x values of much higher than 24.

So, it all comes down to the shape parameter. But how shall we decide on its value? Why, by choosing the one that best fits the data according to Bayes' theorem, of course.

We will outline this procedure in the next post.

Having more "outliers" similar to the maximum makes a true outlier far less likely

Now, what kind of data do we have to determine the shape parameter?

We have the historical data, of course. We have some number of people who are said to have been resurrected in some sense, and each of these people has some amount of evidence associated with their resurrection claim.

We essentially want to "fit" these evidence data into a generalized Pareto distribution, and read off the shape parameter. However, this will be somewhat tricky. We do not have the complete data for all 1e9 reportable deaths throughout human history. We can reasonably assume that the vast majority of them would have essentially zero evidence for a resurrection, but the complete data set would be pretty much impossible to obtain. We don't even have the complete data set just for the outliers - cases like Apollonius or Zalmoxis, where there is a distinctly non-zero level of evidence for a resurrection. Furthermore, the precision on determining the level of evidence is rather poor. All this means that the usual "fit a curve through some kind of x-y scatterplot" approach would not work very well.

However, given that we already know we'll be fitting a generalized Pareto distribution, this is not necessary. We're just looking for the shape parameter, and for that, we merely need to count the number of outliers near the maximum value. Consider the following graph:

This is the same graph as before, in the sense that it just shows the generalized Pareto distribution, scaled so that the probability of x > 1 is 1e-9. Once again, this means that the maximum evidence from 1e9 reportable deaths is likely to appear around x = 1.

However, we now want to focus on how to fit the data. And since the data will have x values less than the maximum, this graph is scaled so that we're focusing to the left of the x = 1 line, instead of the tail to the right.

In particular, note the vast differences in the area under the curve for different shape parameters. The shaded regions represent the probability of finding an "outlier" - a non-Christian resurrection report with at least 20% of the evidence of the maximum report. For instance, the reports of the resurrection of Puhua or Apollonius would be considered an "outlier".

So, let's look at the green curve, with a shape parameter of 20, and a tiny area under the curve. If this were the skeptic's distribution, you'd expect essentially no other outliers. The maximum value would stand by itself, with no other outliers coming anywhere near its value.

Similarly, if the shape parameter is 2, you'd expect perhaps one outlier out of 1e9 samples - one other resurrection report would have at least 20% of the evidence of the maximum.

Lastly, if the shape parameter is 0.2, you'd expect many, many outliers. The probability distribution grows very rapidly as it goes backward from x = 1, and therefore you expect to find many other resurrection reports with a similar level of evidence as the maximum.

So by counting the number of outliers, we can make a determination about the shape parameters.

But... wait a minute. Having more outliers is associated with smaller shape parameters? But didn't smaller shape parameters correspond to a faster-decaying function, and therefore a lower probability for the "skeptic's distribution" generating a Jesus-level of evidence? Wouldn't this lead to the "skeptic's distribution" being less able to explain the evidence for Jesus's resurrection, and therefore make the resurrection more likely?

Are we saying that having MORE non-Christian resurrections reports (like Apollonius or Zalmoxis) make Jesus's resurrection MORE likely?

That is precisely what we are saying. The following analogy may help understand how this could be.
Alice accuses Bob of theft. Bob is known to have come into a sudden possession of $100,000. He is also known to be a gambler. He claims that his sudden fortune came from a lucky night at the card table, but Alice believes that he stole the money - she claims that $100,000 is far too large a sum for Bob to have naturally won through gambling.

Carol takes on this investigation. She looks into Bob's past gambling history, to see it's realistic for him to have won $100,000 in a single night. She finds that, among Bob's past verifiable winnings, there were two nights where Bob won $5,000 and $3000. These are his most remarkable winnings on record, and Carol cannot find any other instances where he won more than $1000 on a single night.

Carol concludes that she does not really have enough information. It could be that Bob plays a card game with an erratic payout scheme, where winning 20 or 30 times more money is not that unusual. Maybe it has some kind of "let it ride" or "double or nothing" mechanism which makes such returns plausible. Or maybe Bob himself is an erratic gambler, and decided to bet a lot more money that one night to win the $100,000. Based on all this, Carol decides to be skeptical of Alice's claim that Bob stole the money. Her own "skeptic's distribution" for how much money Bob can win does not decay quickly enough. There are relatively few outliers near his maximum winnings of $5000, and this suggests that it decays very slowly - meaning that the $5000 cannot be established as a limit to what Bob can win. His theoretical winnings can possibly stretch quite far into the higher values, making it impossible to rule out a $100,000 winning.

But then, Carol has a breakthrough in her investigation. She finds extensive, previously undiscovered records of Bob's gambling winnings, and it shows that Bob has won more than a $1000 on dozens of nights. The maximum that he's won is still $5000, but he's also regularly won thousands of dollars in a single night.

Carol takes this new information into account, and adjust her "skeptic's distribution" for how much Bob can win in a single night. Clearly, Bob's winnings are not erratic; he regularly wins up to about $5000. But this also establishes, with the weight of those repeated winnings, that this is close to the likely upper limit for what he can win in one night.

Carol therefore decides to believe Alice. Her "skeptic's distribution" cannot explain how Bob would naturally win $100,000 in a single night, because it goes against his established pattern of regularly winning up to $5000. She pursues the case further, and eventually convicts Bob of theft.
This is not just a story; it can be mathematically established, and it will be in the future posts. For now, this story just provides the intuitive backing for the mathematical results to come.

So, having more non-Christian reports of a resurrection, with their pathetically low levels of evidence behind them, only make Jesus's resurrection more likely. When skeptics say "don't you know there are numerous other Jesus-like stories of someone dying and resurrecting?", they are only kicking against the goads. The more numerous such cases they come up with, the more firmly it establishes that Jesus really did rise from the dead.

The next post will bring the last several posts together, to fully spec out the program which will compute the complete "skeptic's distribution", from which we can calculate its chances of predicting a Jesus-level of evidence for a resurrection.

The calculation: spell out the details of the program

So then, here is the summary of the basic idea:

We assume that the "skeptic's distribution" will take the form of a generalized Pareto distribution.

We will determine the shape parameter of the distribution by looking at how many "outliers" it has.

A person's resurrection report is considered an "outlier" if it has at least 20% of the evidence compared to the non-Christian resurrection report with the maximum evidence.

The "non-Christian resurrection report with the maximum evidence" is taken to be Krishna or Aristeas, with Apollonius not too far behind. These are taken as having 1/24 of the evidence for the resurrection of Jesus.

Recall that the "some people say... " level of evidence - as per Puhua, Osiris, Zalmoxis, etc. - corresponded to 1/60th of the evidence for Christ's resurrection. This corresponds to 40% of 1/24, meaning that anyone else with the "some people say..." level of evidence would pass the 20% threshold for an "outlier".

We can therefore calculate the relationship between the number of outliers and the likely shape parameters, and thereby calculate the probability for the "skeptic's distribution" naturally generating a Jesus-level of evidence for a resurrection.

Here is the plan for the program:

We will consider shape parameters from 0.02 to 2.1, in increasing intervals of 0.02. That is to say, we will consider shape parameters 0.02, 0.04, 0.06, etc. all the way up to 2.1. Our region of interest will lie in this range.

We will create a generalized Pareto distribution with that shape parameter, then simulate drawing the maximum value of 1e9 samples from that distribution. We will then estimate the number of outliers in that distribution, and the probability of that distribution generating a sample more than 24 times larger than the maximum.

We will do this 10000 times for each value of the shape parameter. This gives us a table of more than a million rows, with each row containing the shape parameter, the number of outliers, and the probability of generating a Jesus-level of evidence.

If we assume equal prior weights for each of the shape parameters, we can consider the final distribution of the shape parameters to be just its distribution from the subset of the table where the number of outliers is equal to the actual, historical value. That is to say, we just have to look at where the theory fits the data, and consider only those theories. This satisfies Bayes' theorem.

Likewise, the probability of the "skeptic's distribution" achieving a Jesus-level of evidence for a resurrection will just be the mean value of that probability from the same subset.

Every assumption and choice made above favors the skeptic's case. Therefore, the probability obtained at the end will be a maximum probability; the skeptic cannot hope for more than that in explaining Jesus's resurrection "naturally".

We will go over the actual computer program in the next post.

Simulation and code: The number of "outliers" decides the case.

(part 39) (WARNING, HTLM)

The list of 50 outliers. This puts the Bayes' factor beyond the prior.

In the previous post, we demonstrated that the likelihood for Christ's resurrection came down to the number of "outliers" we can find in world history - where "outliers" are the other, non-Christian "resurrection" reports with at least a "some people say..." level of evidence behind them. The more such low-evidence cases we find, the more firmly it establishes that low level as the likely maximum for a naturalistically generated resurrection report. The high level of evidence for the resurrection of Jesus Christ then become correspondingly less explicable by naturalistic means, so the resurrection itself becomes more likely.

So, how many such "outliers" can we find? Here is a partial list, including the ones we have already covered:

Aristeas (ancient Greek Poet)
Apollonius of Tyana (Greek philosopher)
Krishna (Hindu god)
Zalmoxis (ancient Greek god)
Osiris (ancient Egyptian god)
Dionysus (ancient Greek god)
Bodhidharma (Buddhist monk)
Puhua (Buddhist monk)
Horus (ancient Egyptian god)
Ba'al (Canaanite god)
Melqart (Phoenician god)
Adonis (ancient Greek god)
Eshmun (Phoenician god)
Tammuz (Sumerian god)
Attis (Phrygian god)
Baldr (Norse god)
Quetzalcoatl (Aztec god)
Izanami-no-Mikoto (goddess in Japanese mythology)
Ishtar (Mesopotamian goddess)
Eurydice (nymph in Greek mythology)
Persephone (ancient Greek goddess)
Asclepius (ancient Greek god/healer)
Hippolytus, son of Theseus (character in Greek mythology)
Cyclopes (character in Greek mythology)
Achilles (ancient Greek hero)
Memnon (ancient Greek hero)
Castor (character in Roman mythology)
Alcmene (character in Greek mythology)
Heracles (ancient Greek hero)
Melicertes (character in Greek mythology)
Romulus (mythic founder of Rome)
Cleitus (character in Greek mythology)
Cycnus, son of Ares (character in Greek mythology)
Cycnus, king of Kolonai (character in Greek mythology)
Cycnus, friend of Phaethon (character in Greek mythology)
Cycnus, son of Apollo (character in Greek mythology)
Odin (Norse god)
Augustus (Roman emperor)
Peregrinus Proteus (Greek philosopher)
Rabbit Boy (character in native American mythology)
Arrow Boy (character in native American mythology)
Man-eagle (character in native American mythology)
Judah the Prince (Jewish rabbi)
Kabir (Indian mystic poet)
He Xiangu (Chinese immortal)
Li Tieguai (Chinese immortal)
Zhang Guolao (Chinese immortal)
People resurrected by Zhongli Quan (associates of a Chinese immortal)
Ye Fashan (Chinese immortal)
Sabbatai Zevi (Jewish rabbi, messiah claimant)

I'm going to stop here - not because I've exhausted such "outliers", but because this is quite enough. The above list contains 50 people who are claimed to have been "resurrected" in some form, with about a "some people say..." level of evidence behind them. And as we saw previously, 50 outliers is enough to reduce the probability for a naturalistic explanation to around 4e-12.

Now, recall that the Christian explanation for the level of evidence for Christ's resurrection is of order unity. Recall also that the prior against the resurrection was set at 1e-11. Combining all this gives a Bayes' factor of greater than 1e11 against a prior of 1e-11 - that is, Jesus's resurrection has better than even odds of having occurred.

Notice that the procedure that got us here outlines the worst case scenario for the resurrection. For example, I stopped the above list of "outliers" 50 only because it was getting tedious to write more. The above list was obtained with just a little bit of Googling, mostly from stories that are readily available online, accessible to a culturally western, English-speaking audience. How many such "outliers" are there in total, throughout all of world history? I would easily imagine it to be in the hundreds, if not thousands.

So, "better than even odds" is the absolute least that can be said for Christ's resurrection. The next post will go back over the procedure that got us to this point - and demonstrate that, in fact, the worst case scenario for the resurrection had been assumed at every point.

Conclusion: The resurrection hypothesis still holds


Chapter 8:
The generosity in the "skeptic's distribution" approach

The more likely adjustments. We were far too generous to the skeptic

Reason 1:


Reason 2:


We have established that the resurrection has, at a minimum, even odds of having taken place. Let us retrace our steps and demonstrate that this is, in fact, the minimum.

Looking back, we see that our first decision was to choose a power law distribution as the "skeptic's distribution". As we mentioned when we made the choice, this is the most pro-skeptical choice we can make that fits the facts. Power law distributions have one of the longest possible tails, which can decay very slowly. They're fully capable of a "black swan" event. Furthermore, they're ubiquitous in human behavior, in that they're naturally generated when an increase in a value depends on the value itself. For this reason, the distributions of personal wealth, city sizes, and website popularity all follow a power law distribution. It's therefore appropriate to use it to model the buildup of evidence through possibilities like conspiracy theories or religious mass delusions.

However, there are excellent reasons to believe that the true "skeptic's distribution" will die off more quickly than a power law distribution, especially when we extend it to 24 times the maximum observed value. You see, few power law distributions can actually extend off to infinity - some external factor will intervene to cut off the distribution at very large values.

Consider city sizes, which we just mentioned. The population of cities follows a power law, and this holds up pretty well as long as we consider populations up to tens of millions of people. However, if we try to extend this out to infinity, the distribution no longer holds. We run into external factors which limit city sizes, such as the total population of humanity or the logistics of city growth in a given geography. For example, the largest city in South Korea is Seoul, with about 10 million people. A city 24 times larger than that would have over 200 million people - much larger than the total population of South Korea, which is only 50 million. Such a South Korean city cannot exist - not because its probability would be too small according to the power law distribution, but because it runs into external factors, like the fact that a city cannot be larger than the country to which it belongs. That is to say, the power law distribution for city sizes is limited, or cut off, at the long tail.

You can imagine similar arguments for personal wealth and website popularity. An individual cannot actually have "all the money in the world", and a website cannot be linked from more websites than the number that actually exist. Likewise, naturalistically generated evidence for resurrection stories cannot follow a power law distribution out to infinity. Other, external factors will cut off or strongly attenuate the probability as such resurrection stories gains more momentum.

For this reason, the true "skeptic's distribution" is almost certainly something that looks like a power law over the actually existing samples, but decays more quickly thereafter. A number of distributions - like a log-normal distribution or a power law with an exponential cutoff - follow this behavior. In each case, these other distributions with their "shorter" tails would help the resurrection case. So adopting a generalized Pareto distribution, which is a genuine power law all the way out to infinity, was therefore the most pro-skeptical choice we could have made.

Next, we considered evenly-spaced shape parameters, in intervals of 0.02, for our distribution. That is to say, we chose a uniform distribution over the shape parameter as our prior. Again, this almost certainly unduly favors skepticism. Consider what such a prior distribution means: the true shape parameter would be 1000 times more likely to be between 1000 and 2000 than between 0 and 1. It would be infinitely more likely to be greater than 1 than to be less than 1. Remember that a larger shape parameter favors the skeptic's case, and we have chosen a prior that favors these larger values. It is only through the weight of the evidence that this prior distribution gets reigned in, but choosing such a biased prior still biases the end results.

The more common and reasonable choice of prior in such circumstances is to consider shape parameters which increase linearly in their logarithms. For example, we may consider shape parameters like 0.01, 0.1, 1, 10, 100, and so on. The idea is that we don't know what the order of magnitude of the shape parameter would be, and therefore consider each order of magnitude equally. Of course, such a prior favors the smaller shape parameters compared to the uniform distribution that we actually used, meaning that it helps the case for the resurrection. So once again, our choice of evenly-spaced shape parameters was the most pro-skeptical choice we could have made.

Next, we considered the maximum value of 1e9 samples drawn from our "skeptic's distribution". That value of 1e9 was chosen as the number of "reportable deaths" in world history. That is, this is the number of deaths that had a chance to be witnessed, documented, or told about in a story. It excludes those deaths where nobody could have made a statement about that death, even if a genuine resurrection took place.

But a moment's reflection shows that this number is too small. Only 1e9 - one billion - "reportable deaths" in world history? More people than that have died just in the last century, and virtually all of these deaths have been "reportable" according to the definition above. Surely a more realistic figure would easily be above 1e10.

This is important, because this sets an upper bound on the probability of generating a Jesus-level resurrection report. A report with the most evidence out of 1e9 samples has a probability of about 1e-9 of being generated. The most evidence out of 1e10 samples would correspondingly have a probability around 1e-10. We then calculate the chances of generating a report with 24 times more evidence.

It's clear that the larger the number of samples, the smaller the probability of generating a report with a level of evidence comparable to the maximal sample. The probability of beating that by a factor of 24 is smaller still. So, the more samples we use, the smaller the probability for the "skeptic's distribution" generating a Jesus-level resurrection report. In other words, using 1e9 as the number of "reportable deaths" was a pro-skeptical choice. The true value is definitely much larger - easily above 1e10. And using this true value would only strengthen the case for the resurrection.

In the next post, we will continue going over more reasons why our previous calculation gives the minimum possible probability value for the resurrection.

Next, consider the factor of 24 that we used, as the ratio between the level of evidence for Jesus's resurrection, and that of the runner-up. This, too, was a very conservative estimate, which favors the skeptic's case.

You'll recall that the runners-up were Aristeas and Krishna, with Apollonius falling not too far behind. In previously reviewing each of these cases, I noted that I was being quite generous in granting them their level of evidence. I felt that there was essentially no evidence, but wanted to express the fact that at least someone had said something - so I somewhat arbitrarily assigned the "some people say..." level of evidence as 1/10th of a single, solid, historic person's sincere testimony. Then, the three people named above then got a multiplier on top of that to account for some additional details.

Of course, this is an overestimate of the "some people say..." level of evidence. Would you believe ten people with "some people say..." stories, over a single person who's giving a sincere, personal testimony? Consider some thought experiments. What would you think of ten people who say, "a friend of a friend whose uncle works for the US government tells me that some people say that the president has been contacted by aliens"? How would that compare with a single witness who sincerely and consistently says "I was there when the aliens contacted the president. It really happened"? If you were a journalist, which source would you cite in your article? If you needed more information, who would you talk to?

Another way to see that the "some people say..." level of evidence is relatively worthless is to observe how common it is. Indeed, this is tied in with the fact that the great number of such evidence throughout history works as evidence FOR the resurrection. We have seen that there is at least 50, and likely hundreds or thousands of such "some people say..." reports for a resurrection. In contrast, none of these non-Christian accounts has even a single, historical person claiming that they personally witnessed the resurrection. This speaks to the relative worth of a single personal testimony over hundreds or thousands of "some people say..." reports.

And remember that this whole time, for the Christian case, we've only been considering the people mentioned in 1 Corinthians 15. This doesn't give many people the full weight their testimonies deserve (John, for example, should be counted more like Peter than just a member of the Twelve), and doesn't take some groups of people (like the women at the cross) into account at all.

All told, the level of evidence for Jesus's resurrection is far greater than 24 times that of the runner up. Using 24 as the factor is a very conservative, pro-skeptical choice.

I've also touched upon the number of "outliers" - the number of resurrection reports with a "some people say..." level of evidence. I've cited 50 such reports, and have used 50 in the calculation as the number of outliers. But as I mentioned, this is a vast underestimate. It comes from a very limited subsample of all the stories in world, reachable by a few minutes with Google in English. The true number of such outliers could easily be in the hundreds or thousands. So this, too, was chosen to favor the skeptical case. In reality, the actual number of outliers would favor the resurrection.

There's still more. Note that, for the "skeptic's distribution", we've integrated out to infinity to get the probability of it explaining the level of evidence for Jesus's resurrection. Strictly speaking, this is an improper way to calculate the Bayesian likelihood. The correct way would be to calculate the probability for the "skeptic's distribution" getting the ACTUAL level of evidence for Jesus's resurrection, rather than calculating the probability of it EXCEEDING that level of evidence. Of course, doing it correctly makes the likelihood smaller, because you're giving up all of that probability out to infinity in the long tail.

The Christian hypothesis must face the same treatment, of course. So its Bayesian likelihood would also drop. But this will not as detrimental as it is for the "skeptic's distribution", because the Christian hypothesis more narrowly focuses the distribution of evidence around the actual value. That is to say, if Jesus did really rise from the dead, it's quite likely for that to have left the amount of evidence that we actually find, while there's good reasons for it to be not that much greater. On the other hand, a long-tailed "skeptic's distribution" would extend on out to infinity - with the consequence that it must pay when the band narrows to the actual amount of evidence we have.

You can argue that the "skeptic's distribution" need not extend on out to infinity - but that just means you're arguing that the "skeptic's distribution" does not follow a power law, instead following another distribution with a stubbier tail. So either the "skeptic's distribution" does worse because it's actually a "stubbier" distribution than a power law, or because it loses its probability mass when we force it to focus around the amount of evidence we actually have in history. In any case, our previous calculation was one that favored the skeptic's case.

Lastly, it's very important to realize that our entire argument about the "skeptic's distribution" only takes the AMOUNT of evidence into account. It argues that no possible effect - not even the ones with a near-total dependence in the evidence (e.g. conspiracy theories) - could falsely generate the amount of evidence for Jesus's resurrection.

Of course, Jesus's resurrection has more than just the sheer AMOUNT of testimonies going for it. We have not yet considered any of the evidence that Christianity has that specifically counters hypotheses like conspiracy theories. These evidence must be considered on top of the mere existence of the numerous testimonies to Jesus's resurrection.

Now, recall that nearly all of the remaining possibility for the skeptic was in crackpot theories like conspiracies. So these evidence against crackpot theories apply nearly their entire weight against the remaining probability for the skeptic. Considering these anti-crackpot evidence would strongly shift the conclusion towards Christianity.

So we see that "even odds" for Jesus's resurrection really is a minimum. It's a value derived by severely discounting and ignoring huge realms of evidence for the resurrection, while granting the skeptic's case every reasonable allowance. The actual odds would be far more favorable towards Christianity.

Next week, we calculate what such "actual odds" may realistically be.

The same kind of calculation for youtube videos and their view counts

(new material)

The simulation and code, with the new adjustments

(part 43) (WARNING, HTML)

Anti-crackpot theory defenses built in to Christianity

(part 37)

Defense 1:


Defense 2:



Conclusion: the resurrection is still certain, even after taking crackpot theories into account

(part 44)

Chapter 9:
One more double check: validation through miracles in other religions


(part 48)

"Something happened" vs. "a miracle happened":

(part 48)


(part 50)

Splitting the Moon

(part 50)

Accounts in Josephus

(part 49)

Honi the Circle-drawer

(part 49)

Eleazar the exorcist

(part 49)


(even with just 10x the level of evidence, with no increase in outliers, it's enough (new material))

Conclusion: this validates our approach. nothing approaches the level of evidence for the resurrection



Challenge and Conclusion

Chapter 10:
The final challenge: replicate the results

Replicate the results part 1

(part 45)

Replicate the results part 2

(part 46)

Chapter 11:
Conclusion and epilogue


(part 51)


(part 52)

Rules for formatting, organization:


Part (double space at end)


Each chapter ends with a conclusion. Part 4 is the conclusion for the series.

everything uses default fonts, all heading emphasis is done through centering, whitespace, bold, etc.

You may next want to read:
The Gospel: the central message of Christianity (part 1)
How is God related to all other fields of study?
Another post, from the table of contents