September 13, 2016

(Continued from the previous post)

Now, I acknowledge that your "license plate effect" is in fact real - that 1e8 can be split between the "license plate effect" and the remaining "human honesty factor". But for the examples that I provided, I disagree that the split between these factors is as extreme as 1e6 to 1e2. I mean, if you've won the lottery, there generally aren't 1e6 other things that you could have chosen to lie about that would be equally interesting.

But more importantly, the exact split doesn't matter. The examples from which I calculated that 1e8 value were all specifically chosen to be similar to the disciple's testimony for Christ's resurrection. They are all special, interesting, positive claims - indeed in each case they're probably one of the most interesting things that the person could talk about. So, as long as this similarity holds, my full value - both the "license plate effect" and the "human honesty factor" - is applicable to the disciple's testimony. As Bayes' rule says, posterior odds is prior odds times Bayes's factor. And that Bayes' factor doesn't care whether certain parts of it were from certain effects. The entire Bayes' factor applies, as long as it was calculated for a similar situation. This is why I'm not too concerned about, say, your counterexample of "[naively slapping] 8 orders of magnitude on a 1:1 odds proposition": because that does not correspond to the scenario that the disciples faced.

Now, this "license plate effect" and the "human honesty factor" does interact in interesting ways in cases of multiple testimonies. I'm not sure if this is what you were getting at when you said that "if I'm right about the 8 = 6 + 2 split, you can only discount that 6 once". But I believe something like that can happen, depending on the degree of dependence between the multiple testimonies.

This is how it would work out (using your numbers): if Peter testifies that Jesus rose from the dead, then you should give his testimony the full 1e8 = 1e6 * 1e2 Bayes factor - exactly as it worked out in my numerous examples. But if you then turn to John and ask "hey, is Peter telling the truth?" and John answers "yes", John should only get the 1e2, because he has now been put in a different kind of situation than Peter - more akin to answering an 1:1 odds proposition.

But if you then have someone completely random burst into the room afterwards - let's say Paul - who says "guys! Jesus rose from the dead!", then that testimony should again get the full 1e8 Bayes' factor, because it was made independently from the other two.

So this again turns into a question of independence. Fortunately, I do think there is a strong case to be made for independence among the three named witnesses I used - Peter, James, and Paul. I mean, you yourself gave a 1e6 factor to Paul's conversion because it was so unexpected that an enemy of Christianity would have such a drastic turnaround by the encounter with the risen Christ. That's a factor you gave out on top of the fact that people claimed to have seen the risen Christ, as an expression of the strong anti-correlation (beyond mere independence) you'd expect between Paul's testimony and the other's. That was an additional factor on top of what's in my calculations, and it amply covers any possible dependence between Peter and James. As for the remaining testimonies in 1 Corinthians 15, I did severely discount them to account for dependence. So I'm quite comfortable with my 1e54 value, with the aforementioned caveat that we're not yet considering things like conspiracy theories.

Now, as I said, I disagree with your 1e6:1e2 split for the "license plate effect", and the implication that this effect mainly serves to weaken the witness testimonies. I think that its more important function, by far, is to immensely strengthen the testimonies to which it applies. It works to "cancel out any amount of low prior probability", in your words. Or to empower the testimony with a Bayes' factor of something like 1e120, in my example of recording a chess game.

So I'm very glad you mentioned the effect, because it was somewhat foggy in my own mind, and it allows us to do things like justify the remainder of the stories in the Gospel accounts. These other stories just get filed under "more details" once the resurrection is accepted, whereas there's no way to cover the prior on all those stories on just a Bayes' factor of 1e8.

But at the end of this discussion on the strength of a single testimony, I still pretty much stand by my 1e8 number. I think 1e7 is also quite reasonable, and it can drop to something like 1e6 in circumstances where the possibility of lying is distinctly real (e.g. claiming to an aid agency that a loved one died on 9/11, or when calculating how many times you've been lied to). Maybe I could be convinced to use 1e6 as a lower bound, instead of 1e8 as a likely value. In any case it would not really affect my series, as I've said that a human testimony is "within a couple of orders of magnitude of my answer", and even the lower value is large enough to overwhelm any possible prior against the resurrection.

As one more confirmation of that 1e8 number, take a look at this video - it show's a woman's reaction to an acquaintance claiming to have won the lottery. Now, did that woman seem like a gullible idiot to you? I didn't feel that way. She starts off quite skeptical, but not dismissively skeptical. You can then see the man's sincerity working on her. Her degree of belief is clearly somewhat close to even odds right before the numbers are confirmed. I think her overall reaction is pretty rational. Now, there are some small differences between the video and my examples. For instance, she knows that there's a winner out there, and the man making the claim is already an acquaintance - but on the other hand, this result is achieved with little effort on the man's part, taking only minutes of insistence. The man being an acquaintance also reduces the "licenses plate effect". On the whole, you can see her mind being pulled through a Bayes' factor of something like 1e6 within mere minutes, in good accord with rationality, in a situation pretty similar to what I described in my examples. So 1e8 for something like the disciple's testimony about the resurrection is quite reasonable, and remains the best value to use.

(To be continued in the next post)

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