April 12, 2016

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

Another post, from the table of contents

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