Bayesian evaluation for the likelihood of Christ's resurrection (Part 6)

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.

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 5)

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.

You may next want to read:
How physics fits within Christianity (part 2)
Science as evidence for Christianity (Summary and Conclusion)
Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection (Part 4)

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.

You may next want to read:
Sherlock Bayes, logical detective: a murder mystery game
Key principles in interpreting the Bible
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Bayesian evaluation for the likelihood of Christ's resurrection (Part 3)

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 entirity. 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.

You may next want to read:
Sherlock Bayes, logical detective: a murder mystery game
How to think about the future (Part 1)
Another post, from the table of contents