Bayesian evaluation for the likelihood of Christ's resurrection (Part 19)

Fortunately, skeptics of Christ's resurrection often do some of the early leg work for us, in that they compile lists of purported people who have been said to be like Christ for one reason or another. We'll look at a representative sample from such lists.

First, let us consider Apollonius of Tyana, who is sometimes compared to Christ because they were both philosopher/preachers in first century Rome, to whom miraculous powers are attributed. Wikipedia has a list of similarities between Jesus and Apollonius, which includes a wondrous birth, the ability to heal the sick and raise the dead, a condemnation by Rome, and an ascension into heaven. That sounds pretty similar, no? So how does the evidence for Apollonius's "resurrection" hold up?

Pathetically. Most of the information on Apollonius comes from Philostratus, who was paid to write a biography of Apollonius well over a hundred years after Apollonius's death, and after Christianity was already a thing. This biography only "implies" that Apollonius underwent heavenly assumption. Furthermore, the chief primary source for this biography is one Damis, a disciple of Apollonius, who is unknown outside of this biography. And to top it off, Philostratus specifically writes that Damis had not recorded anything about Apollonius's death. The stories of his death and supposed heavenly assumption are in a part of the biography that are filled with 'some say this, some say that' stories, which, by the author's own admission, he wrote because he felt that his story needed to have a natural ending.

So, the evidence for Apollonius's "resurrection" comes down to one author, who wrote more than a hundred years after the event, who says that he's getting his information second-hand from a Damis that nobody else has heard of, who then says that the "resurrection" bit - which is only implied - doesn't even come from Damis.

Compare that to the evidence for Christ's resurrection, in the form of the testimony of his disciples. 1 Corinthians 15 was written within a couple decades of the event, and it contains a creed that was formulated mere years after the resurrection. We have the personal, first-hand testimonies of the people who have seen the risen Christ. They clearly say that it really happened, and that it transformed their own lives. Each of these disciples appear in multiple other sources, and bear the same witness about Christ's resurrection in those sources.

So... now I'm suppose to compare the strength of the evidence between these two? Well, let's see. Remember our previous criteria, about what it would take to "match" 1/6th of the evidence in 1 Corinthians 15. Can we say that maybe that Damis's testimony about Apollonius's resurrection matches the testimony of Peter, James, or Paul? Well, no. Damis never made that testimony, nor is he anything like those three individuals on the quality of historical information we have on him. So then, all that's left as evidence is "some say that Apollonius rose from the dead", stated more than a hundred years after the fact?

That is essentially no evidence. But since I have to give a numerical estimate, I would be generous and say that Damis's "testimony" counts as an order of magnitude less than that of Peter, James, or Paul. I will also generously grant the "some say..." part of the story as being a order of magnitude less than that of the 500 witnesses that Paul mentions. So, that comes to:

1/6 (matching a single element in 1 Corinthians 15)
× 1/10 (an order of magnitude less)
× 2 (two such instances),
= about 1/30th of the evidence that we have for Christ's resurrection.

We will continue with other personages in the next post.

You may next want to read:
What is "evidence"? What counts as evidence for a certain position?
Christianity and falsifiability
Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection (Part 18)

We are interested in quantifying the Bayes' factor for the testimonies concerning Christ's resurrection. We will do so by comparing the level of evidence for Christ's resurrection against the level of evidence we find in world history through a naturalistic process and outlook. Here is how the procedure will work:

As a conservative estimate, let us say that there have been 1e9 reportable, naturalistic deaths throughout world history. Then, a Bayes' factor of 1 would correspond to finding that all 1e9 of those deaths also resulted in resurrection stories, each as well-evidenced as Christ's resurrection.

A Bayes's factor of 1e9 would mean that there is still one other naturalistic death which resulted in a story on par with Christ's resurrection.

A Bayes' factor of 1e18 would mean that in all the rest of the world, among the 1e9 reportable, naturalistic deaths, there may be some deaths which has a resurrection story - but they would only have at most about half the evidence that Christ's resurrection has. The rationale is that if achieving half-evidence has a 1e-9 chance of happening, then the chance of that happening twice for the same event would take 1e9 squared, or 1e18, number of events.

By the same reasoning, a Bayes' factor of 1e54 would imply that the closest event to Christ's resurrection would have about 1/6th the evidence that Jesus's resurrection has.

So then, what are these evidences for Christ's resurrection? And what would 1/6th of that evidence look like? For our current purposes, we're using the testimonies enumerated in 1 Corinthians 15 - that's the passage from which we calculated our value of 1e54 for the Bayes' factor. In getting that number, I specifically accounted for the earnest, insistent, sincere personal testimonies of Peter, James, and Paul, as well as the group testimonies of the other apostles and the 500 disciples.

That's five distinct sets of testimonies. Not all of them were given equal weight in my calculations - the least of these were only responsible for 1e8 out of 1e54 in terms of Bayes' factors, or about 1/7 of the total weight of evidence. So, let's be conservative here (as I've always been throughout this entire series), and say that if any other non-Christian story about a resurrection matches any one of the five sets of testimonies in Acts 15, that would count as achieving 1/6th the evidence of Christ's resurrection.

What would it take to "match" one of these sets of testimonies? Well, the idea here is that the overall quality of the evidence should be on par with the evidence we have from the testimonies mentioned in 1 Corinthians 15. So:

To match Peter, James, or Paul's testimonies, we will require an earnest, insistent, and personal testimony by a single named individual, whom we can historically locate with great precision. We will also require that a good amount of information is available about the life of the person giving the testimony.

To match the testimonies of the other apostles would require an earnest, insistent, and personal testimony by a group of people, most of whom are named and known in history, who can be located with good precision.

To match the testimonies of the group of 500 disciples would require the earnest personal testimonies from a large, specific group of people, who can be well-located in history. They don't have to be named, and they don't have to be insistent in their testimony. But they should be well-defined enough that many of them could be individually pointed out by a well-known historical personage like Paul.

That will be our methodology. Starting in the next post, we will examine the claims where someone was said to have been raised from the dead, like Jesus. We will look at these claims and assign numerical values to the amount of evidence they have, using the criteria specified above. We will then assign a Bayes' factor for the evidence for Jesus's resurrection, based on how it measures up against the evidence for these other supposed resurrections.

You may next want to read:
Sherlock Bayes, logical detective: a murder mystery game
Orthodoxy vs. living out the Gospel: which is more important?
Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection (Part 17)

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.

You may next want to read:
How is "light" used in the Bible, particularly in the creation story?
Come visit my church
Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection (Part 16)

Let us summarize our investigation into the Bayes' factor for a human testimony.

At the beginning of this series, we began by examining our gut feelings on how much credit we would give to someone who claims to have won the lottery or been struck by lightning. From this initial calculation, using just some intuition, we got a variety of numbers for the Bayes' factor, ranging around 1e7 to 1e9. The number we ended up using, 1e8, started from these calculations.

That's a good start, but it needs empirical backing. The first natural experiment we used to verify this number was the case of the people who lied about winning the 1.6 billion dollar Powerball lottery. The result from this calculation was about as good as it could possibly be expected; 1e8 really turned out to be the correct order of magnitude for the Bayes' factor, when someone claimed that they had they had won the lottery.

We then investigated the case of someone missing an appointment due to a car accident. The claim of a car accident on a specific day turned out to have a Bayes' factor of 1e5 as a lower bound, while its true value was estimated to be around 1e7.

We next investigated the tragic story of a young woman dying in a car accident, and her mother committing suicide when she heard the news. The testimony of the person who related this story was calculated to have a Bayes' factor of 1e8 as a lower bound, while its true value was estimated to be around 1e11.

For the claims of being in Harvard's physics PhD program, the Bayes' factor was found to have a lower bound of 1e7, with no estimate for the most likely value. And for the case of people claiming to have lost a close loved one in the 9/11 attacks, the Bayes' factor turned out to be about 1e6, despite the fact that there was cold, hard cash to be won as a strong temptation to lie.

So, the Bayes' factor for an earnest, sincere, insistent personal testimony really is about 1e8, and this is born out by multiple lines of thought, and verified by multiple cases of empirical inquiry.

It is important to note that these examples were merely the first ones that came to my mind which I could also get good numbers for. There is no selection bias here. There is not a set of other examples which I chose not to use because they did not prove my point or suit my purpose. In fact, I encourage you to come up with your own examples through which you can compute the Bayes' factor of a human testimony. Compare your answer with mine, and independently verify my values.

It is also important to acknowledge that there is variance in the Bayes' factors. That 1e8 is a typical value, and it will naturally change when we put conditions on it. For example, the relatively low value of 1e6 for people claiming to have lost loved ones in the 9/11 attack can be attributed to the possibility of dishonest gain through fraud. On the other hand, the high value of 1e11 was obtained for a friend telling me an unlikely story, and its greater Bayes' factor can perhaps be attributed to the fact that I was friends with this person. It seems that such considerations can shift the Bayes' factor by a couple orders of magnitude, although it's also worth noting that there are also cases like reporting the moves of a chess game, where the Bayes' factor exceeds 1e120.

Remember, a Bayes' factor of 1e4 (and its corresponding discount for lesser testimonies) still gives a 99.999% chance that Jesus really rose from the dead, even starting from the ridiculously low prior odds of 1e-22. So for our purposes, the important thing in these investigations is that the Bayes' factor always exceeded 1e4 - even for the minimum estimates in each case. Furthermore, the minimums always exceeded 1e4 by more than a couple of orders of magnitude - which is the above-mentioned typical variation in the Bayes' factor under different conditions. Meaning, EVEN IF you believe that the disciples had a good reason to be deceptive or delusional, there's STILL enough evidence in their weakened testimonies to conclude that Jesus did really rise from the dead. That's how strong the case for the resurrection is.

Of course, we've already covered the issue of how the disciple's testimonies may vary from the "typical" testimony. We've seen that in every way, their testimonies are in fact stronger than the testimonies of a "typical" person. The variation in their Bayes' factor from 1e8 is therefore going to be towards larger numbers, like 1e11 or 1e120, rather than towards 1e4. 1e8 is an underestimate of the Bayes' factor, and therefore 1e32 is also an underestimate for the odds for Christ's resurrection. Jesus almost certainly rose from the dead.

In the next post, we'll tackle this question of the Bayes' factor for the resurrection testimonies from a different angle.

You may next want to read:
How physics fits within Christianity (part 1)
Human laws, natural laws, and the Fourth of July
Another post, from the table of contents