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)

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.

(part 1, 3)

(part 4)

(text)

(part 12)

(parts 25, 26, new material)

(part 27, 28)

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

(part 13)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.

(part 13)

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.

(part 14)

(part 15)

(new material)

(part 30)

(part 26)

(text)

(part 16)

(part 17)

(part 18)

(part 19)

(part 20)

(part 21)

(part 21, new material)

(part 21)

(part 21)

(part 22)

(part 23)

(part 24)

(part 29, 30)

(part 5)

(part 6)

(part 7)

(part 8)

(part 10)

(text)

(part 9)

(part 10)

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

(text)

(addressing "disciples stole the body" like theories)

(part 11, 30)

(part 31)

(text)

(part 34)

(part 35)

(part 36)

(part 37)

(part 38)

(part 39)

(part 40)

(text)

(part 41, 42)

(text)

(text)

etc.

(new material)

(part 43)

(part 37)

(text)

(text)

etc.

(part 44)

(part 48)

(part 48)

(part 50)

(part 50)

(part 49)

(part 49)

(part 49)

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

(text)

(part 45)

(part 46)

(part 51)

Rules for formatting, organization:

hierarchy:

Part (double space at end)

Chapter

heading

sub-heading

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

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)

**PART I:**

**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**(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.**(text)

**Chapter 2:**

**Double checking the strength of a human testimony**

**The rationale for double checking**(part 12)

**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:**(part 13)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:**(part 13)

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:**(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.**(text)

**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**

**Rational: 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, new material)

**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...**(text)

**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, addressing "disciples stole the body" rumor)

**Conclusion: The Bayesian argument takes it all into account**(text)

**PART III:**

**Addressing all possible alternatives**

**Chapter 6:**

**Time to address the crackpot theories**

**Why we now turn to crackpot theories**(addressing "disciples stole the body" like theories)

(part 11, 30)

**Examining crackpot theories, in general**(part 31)

**Conclusion: we now turn to examining even such theories**(text)

**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: The resurrection hypothesis still holds**(text)

**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)

**Reason 1:**(text)

**Reason 2:**(text)

etc.

**The same kind of calculation for youtube videos and their view counts**(new material)

**The simulation and code, with the new adjustments**(part 43)

**Anti-crackpot theory defenses built in to Christianity**(part 37)

**Defense 1:**(text)

**Defense 2:**(text)

etc.

**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**

**Vespasian**(part 48)

**"Something happened" vs. "a miracle happened":**(part 48)

**Ichadon**(part 50)

**Splitting the Moon**(part 50)

**Accounts in Josephus**(part 49)

**Honi the Circle-drawer**(part 49)

**Eleazar the exorcist**(part 49)

**Recalculation:**(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**(text)

**PART IV:**

**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**

**Conclusion**(part 51)

**Epilogue****(part 52)**

Rules for formatting, organization:

hierarchy:

Part (double space at end)

Chapter

heading

sub-heading

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

### Bayesian evaluation for the likelihood of Christ's resurrection (Part 52)

This is a kind of epilogue for my series on Jesus's resurrection. The series is now concluded - or, at least, the first phase is done.

What happens next? Longtime readers of my blog know that I generally consolidate long, multi-part posts into a single post in the end. This gives me a way to present the whole series as an unified, definitive work, and allows for easy editing and upkeep. I will now proceed to do that for the series on the resurrection.

But this resurrection series is by far the longest series on my blog. It represents a year's worth of work, totalling tens of thousands of words (long enough for it to be a decent length book) in 52 parts. It will take some time to consolidate and edit. In particular, this process will take significantly longer than one week, which is how long it's taken for my other series. I ask for your continued patience, support, and feedback during this time. I will still update the blog weekly - it's just that the new posts will just be a summary of the edits, with a link back to the changing, updated post.

The first part of this series was posted March 21, 2016 - The week of Easter. When I started writing it, I only had the idea that you can actually calculate a value - a Bayes' factor - for a human testimony. In a demonstration of my poor long-term planning skills, I initially just wanted to write one post for Easter, about the likelihood of the resurrection based on the Bayes' factor of the disciple's testimonies. But the ideas kept coming, they all required thorough explanations, and the posts just kept on writing themselves.

I furthermore want to point out that throughout the writing process for this whole series, I've never thrown out bad results or concealed disadvantageous conclusions. Every time a new idea came to me - every time there was a new way to test the veracity of the resurrection - I explored it, quantified the essential thoughts, thoroughly performed all the necessary calculations, and presented the results. There were some branches of thought that did not make it into the series, but these were all because the initial idea was not mathematically workable, or because they ended up being redundant to the thoughts that did make it into the series. Again, there was never a single instance where I reached a conclusion against the resurrection, and decided to hide or ignore it. There is no selection or confirmation bias here. The resurrection was validated each and every time.

So after a little over a year, we come to another Easter, and this milestone in the series. It's been a good, productive year, and I'm glad to wrap up this first phase of this work. Happy Easter to you all - Christ is risen indeed!

You may next want to read:

"Simon, son of John, do you love me?"

The Gospel: the central message of Christianity (part 1)

Another post, from the table of contents

What happens next? Longtime readers of my blog know that I generally consolidate long, multi-part posts into a single post in the end. This gives me a way to present the whole series as an unified, definitive work, and allows for easy editing and upkeep. I will now proceed to do that for the series on the resurrection.

But this resurrection series is by far the longest series on my blog. It represents a year's worth of work, totalling tens of thousands of words (long enough for it to be a decent length book) in 52 parts. It will take some time to consolidate and edit. In particular, this process will take significantly longer than one week, which is how long it's taken for my other series. I ask for your continued patience, support, and feedback during this time. I will still update the blog weekly - it's just that the new posts will just be a summary of the edits, with a link back to the changing, updated post.

The first part of this series was posted March 21, 2016 - The week of Easter. When I started writing it, I only had the idea that you can actually calculate a value - a Bayes' factor - for a human testimony. In a demonstration of my poor long-term planning skills, I initially just wanted to write one post for Easter, about the likelihood of the resurrection based on the Bayes' factor of the disciple's testimonies. But the ideas kept coming, they all required thorough explanations, and the posts just kept on writing themselves.

I furthermore want to point out that throughout the writing process for this whole series, I've never thrown out bad results or concealed disadvantageous conclusions. Every time a new idea came to me - every time there was a new way to test the veracity of the resurrection - I explored it, quantified the essential thoughts, thoroughly performed all the necessary calculations, and presented the results. There were some branches of thought that did not make it into the series, but these were all because the initial idea was not mathematically workable, or because they ended up being redundant to the thoughts that did make it into the series. Again, there was never a single instance where I reached a conclusion against the resurrection, and decided to hide or ignore it. There is no selection or confirmation bias here. The resurrection was validated each and every time.

So after a little over a year, we come to another Easter, and this milestone in the series. It's been a good, productive year, and I'm glad to wrap up this first phase of this work. Happy Easter to you all - Christ is risen indeed!

You may next want to read:

"Simon, son of John, do you love me?"

The Gospel: the central message of Christianity (part 1)

Another post, from the table of contents

### Bayesian evaluation for the likelihood of Christ's resurrection (Part 51)

At long last, we can summarize this entire series.

First, we calculated the prior odds for the resurrection of Jesus Christ. This prior cannot be zero. That would violate one of the fundamental tenets of Bayesian thinking, and it is not empirically justified, since we have not observed an infinite number of people who did not come back from the dead. Instead, empiricism demands that this prior be about the same as the reciprocal of the total number of non-resurrecting people we have observed, even if we have observed zero resurrections. Rather generously, this could be placed at 1e-11 - roughly corresponding to observing the non-resurrection of the total number of humans that have ever lived.

Second, we calculated the Bayes' factor for an earnest, insistent human testimony. Human testimony has value. This is not just an opinion or a hypothesis: human testimony must have value because your odds for an event actually changes when someone makes a testimony. Therefore, it must have a Bayes' factor. Thus the value of a human testimony can be calculated on a mathematical and empirical footing. As it turns out, for a testimony like the ones concerning Jesus's resurrection, the Bayes' factor is about 1e8. This is validated by multiple empirical observations, natural experiments, and thought experiments. There are several ways to modify this value depending on the exact nature of the testimony, which include things like dependency factors, an incentive to lie, and the "license plate effect". All of these can be understood and were taken into account.

We next evaluated the amount of evidence for Jesus's resurrection. Just from a stripped-down version of the testimonies summarized in 1 Corinthians 15, we saw that Peter, James, Paul, the 12 disciples, and a crowd numbering more than 500 all testified to Jesus's resurrection. Applying the Bayes' factor calculated above - with the appropriate modifications - to this set of evidence gave an enormous Bayes' factor, easily enough to completely overwhelm the prior odds of 1e-11 against the resurrection. We therefore concluded that Jesus almost certainly rose from the dead.

This calculation was then double checked against other historical reports of a resurrection. By comparing with the non-Christian resurrection reports, we saw that the level of evidence behind Jesus's resurrection is a clear outlier, to an absolutely absurd degree. This comparison therefore validated our earlier conclusion that Jesus rose from the dead.

Furthermore, because of the nature of this calculation, its conclusion is immune from many of the common skeptical arguments against the resurrection. The various possibilities - all the likely ways that the testimonies could have been wrong - have been already taken into account. No amount of speculation about how the resurrection reports could have been generated by naturalistic chance has any effect on the conclusion. We don't need to play 'what-if whack-a-mole' against the skeptic's speculations. This is a Bayesian argument. Speculations do absolutely nothing against it. Only evidence moves the odds.

However, because the Bayes' factor for Jesus's resurrection is so large, we then had to start worrying about crackpot theories - conspiracies, vivid mass hallucinations, alien interference, and the like. At the level of certainty which was implied by our calculation, we had to take even such things into account. This required a recalculation, specifically to take into account the near-total interdependency of evidence implied by such theories.

Fortunately, we had the historical data about other resurrection reports. This allowed us to explicitly construct the "skeptic's distribution", which is the probability distribution that generates the other historical resurrection reports through naturalistic means. This construction explicitly took into account the possibility of crackpot theories. This is the distribution that the skeptic must use, if they are to hold on to empiricism and naturalism - for this distribution incorporates the empirical, historical results of all such crackpot theories at the rate which actually occurred throughout history, and furthermore continues the distribution beyond the empirical end of the distribution using an exceedingly generous set of assumptions for the skeptic.

But even after taking even the crackpot theories into account, with a set of highly favorable assumptions for the skeptic, we saw that the Bayes' factor for the testimonies for Jesus's resurrection still enough to amply overpower the 1e-11 prior odds.

Furthermore, this was only after taking into account the raw amount of evidence summarized in 1 Corinthians 15. We did not take into account all the other people who testified to the resurrection in the New Testament (e.g. the women), and we did not take into account any of the strong pro-independence, anti-conspiracy properties of the testimonies (e.g. Paul's conversion). Including such factors would greatly strengthen an already nearly certain conclusion.

Therefore, this recalculation affirmed our previous conclusion: Jesus almost certainly rose from the dead.

As yet another double-check of our methodology, we tackled a number of other, non-Christian, non-resurrection miracle stories, using the same methodology. We reached the same conclusions that a skeptic would want us to reach - that probably nothing supernatural took place in these cases. Because this double-check reached the same conclusions as the skeptic, the skeptic must therefore count this as an additional validation of the methodology.

Furthermore, our methodology allows the Christian to say that the resurrection (and other Christian miracles) almost certainly took place, while at the same time consistently saying that the non-Christian miracles were probably not actually supernatural. The resurrection just has that much evidence behind it compared to non-Christian miracle stories.

But if anyone is still not convinced of Jesus's resurrection, I have the following challenge for them: naturalistically replicate the resurrection reports. Using the same means that were available to Jesus and his disciples - no political power, no great wealth, no modern science, etc. - generate multiple, detailed, independent, earnest, insistent personal testimonies from a great number of diverse types of people, unanimously testifying to a singular resurrection event. Furthermore, this must be achieved in spite of deadly persecution, in a fractious movement with no central control, along with a host of other difficulties and conditions. If you still doubt Jesus's resurrection, the scientific method demands that you take up this challenge.

Alternatively, you can follow the logic of the methodology outlined above, which is based on mathematics and empirical data, has been validated and double-checked multiple times, and gives the correct answer in all cases where the answer could be agreed upon. Short of embracing epistemological obliteration, you must accept its conclusion: Jesus almost certainly rose from the dead.

The next post is something a little different - a bit of an epilogue, and the future of this series.

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

First, we calculated the prior odds for the resurrection of Jesus Christ. This prior cannot be zero. That would violate one of the fundamental tenets of Bayesian thinking, and it is not empirically justified, since we have not observed an infinite number of people who did not come back from the dead. Instead, empiricism demands that this prior be about the same as the reciprocal of the total number of non-resurrecting people we have observed, even if we have observed zero resurrections. Rather generously, this could be placed at 1e-11 - roughly corresponding to observing the non-resurrection of the total number of humans that have ever lived.

Second, we calculated the Bayes' factor for an earnest, insistent human testimony. Human testimony has value. This is not just an opinion or a hypothesis: human testimony must have value because your odds for an event actually changes when someone makes a testimony. Therefore, it must have a Bayes' factor. Thus the value of a human testimony can be calculated on a mathematical and empirical footing. As it turns out, for a testimony like the ones concerning Jesus's resurrection, the Bayes' factor is about 1e8. This is validated by multiple empirical observations, natural experiments, and thought experiments. There are several ways to modify this value depending on the exact nature of the testimony, which include things like dependency factors, an incentive to lie, and the "license plate effect". All of these can be understood and were taken into account.

We next evaluated the amount of evidence for Jesus's resurrection. Just from a stripped-down version of the testimonies summarized in 1 Corinthians 15, we saw that Peter, James, Paul, the 12 disciples, and a crowd numbering more than 500 all testified to Jesus's resurrection. Applying the Bayes' factor calculated above - with the appropriate modifications - to this set of evidence gave an enormous Bayes' factor, easily enough to completely overwhelm the prior odds of 1e-11 against the resurrection. We therefore concluded that Jesus almost certainly rose from the dead.

This calculation was then double checked against other historical reports of a resurrection. By comparing with the non-Christian resurrection reports, we saw that the level of evidence behind Jesus's resurrection is a clear outlier, to an absolutely absurd degree. This comparison therefore validated our earlier conclusion that Jesus rose from the dead.

Furthermore, because of the nature of this calculation, its conclusion is immune from many of the common skeptical arguments against the resurrection. The various possibilities - all the likely ways that the testimonies could have been wrong - have been already taken into account. No amount of speculation about how the resurrection reports could have been generated by naturalistic chance has any effect on the conclusion. We don't need to play 'what-if whack-a-mole' against the skeptic's speculations. This is a Bayesian argument. Speculations do absolutely nothing against it. Only evidence moves the odds.

However, because the Bayes' factor for Jesus's resurrection is so large, we then had to start worrying about crackpot theories - conspiracies, vivid mass hallucinations, alien interference, and the like. At the level of certainty which was implied by our calculation, we had to take even such things into account. This required a recalculation, specifically to take into account the near-total interdependency of evidence implied by such theories.

Fortunately, we had the historical data about other resurrection reports. This allowed us to explicitly construct the "skeptic's distribution", which is the probability distribution that generates the other historical resurrection reports through naturalistic means. This construction explicitly took into account the possibility of crackpot theories. This is the distribution that the skeptic must use, if they are to hold on to empiricism and naturalism - for this distribution incorporates the empirical, historical results of all such crackpot theories at the rate which actually occurred throughout history, and furthermore continues the distribution beyond the empirical end of the distribution using an exceedingly generous set of assumptions for the skeptic.

But even after taking even the crackpot theories into account, with a set of highly favorable assumptions for the skeptic, we saw that the Bayes' factor for the testimonies for Jesus's resurrection still enough to amply overpower the 1e-11 prior odds.

Furthermore, this was only after taking into account the raw amount of evidence summarized in 1 Corinthians 15. We did not take into account all the other people who testified to the resurrection in the New Testament (e.g. the women), and we did not take into account any of the strong pro-independence, anti-conspiracy properties of the testimonies (e.g. Paul's conversion). Including such factors would greatly strengthen an already nearly certain conclusion.

Therefore, this recalculation affirmed our previous conclusion: Jesus almost certainly rose from the dead.

As yet another double-check of our methodology, we tackled a number of other, non-Christian, non-resurrection miracle stories, using the same methodology. We reached the same conclusions that a skeptic would want us to reach - that probably nothing supernatural took place in these cases. Because this double-check reached the same conclusions as the skeptic, the skeptic must therefore count this as an additional validation of the methodology.

Furthermore, our methodology allows the Christian to say that the resurrection (and other Christian miracles) almost certainly took place, while at the same time consistently saying that the non-Christian miracles were probably not actually supernatural. The resurrection just has that much evidence behind it compared to non-Christian miracle stories.

But if anyone is still not convinced of Jesus's resurrection, I have the following challenge for them: naturalistically replicate the resurrection reports. Using the same means that were available to Jesus and his disciples - no political power, no great wealth, no modern science, etc. - generate multiple, detailed, independent, earnest, insistent personal testimonies from a great number of diverse types of people, unanimously testifying to a singular resurrection event. Furthermore, this must be achieved in spite of deadly persecution, in a fractious movement with no central control, along with a host of other difficulties and conditions. If you still doubt Jesus's resurrection, the scientific method demands that you take up this challenge.

Alternatively, you can follow the logic of the methodology outlined above, which is based on mathematics and empirical data, has been validated and double-checked multiple times, and gives the correct answer in all cases where the answer could be agreed upon. Short of embracing epistemological obliteration, you must accept its conclusion: Jesus almost certainly rose from the dead.

The next post is something a little different - a bit of an epilogue, and the future of this series.

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