Bayesian evaluation for the likelihood of Christ's resurrection (Part 15)

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

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

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

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

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

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

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

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

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

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

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 14)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 13)

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

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.

Now, what about the Bayes' factor in the following scenario?

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.

There are more calculations to come in the next week.

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 12)

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

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

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

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

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

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

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

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