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

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

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

Well, can you demonstrate that empirically?

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

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

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

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

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

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

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

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

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

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

We'll begin this calculation next week.