July 4, 2016

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

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