(Continued from the previous post)
Here's a few more assorted thoughts:
I still think that you're too afraid of large odds. For example, my gut feeling is that 99.99% is far too small a limit on how certain we can be in history. I mean, we can make meaningful, almost empirical statements about everyone who existed. That should tell you that odds of something like 1e11 is not extraordinary. In fact, if we are quite certain in our statement about everyone who lived, we can add a few more orders of magnitude. I still mostly stand by my statement that a probability is not "too large" unless its log is "too large", although I have adjusted this "too large" value downwards somewhat. Or maybe a probability is not "too large" unless you can no longer meaningfully get at it empirically.
On a related note, you see that using the skeptical prior value of 1e-22 for the resurrection is not all just showmanship. Clearly, a prior value of 1e-11 is justified, in that Christianity is essentially claiming that Jesus was unique on all of world history. You'll probably want another 2 orders of magnitude on that to future-proof the claim for the possible increases in the human population. You'll then want another 4 orders to at least reach the 99.99% for "certain within limits of history" mark. That already brings us to 1e17 as the necessary Bayes' factor - which, incidentally, seems to be about what you come up with in your own summary. I'm just requiring another 5 orders of magnitude on top of that to really put the nail in the coffin, and to cover any unseen contingencies. You can call that "just showmanship" if you'd like, but it serves the honest purpose for which I have constructed it in my series - a prior smaller than any that a skeptic can justifiably ask for, which can still be overcome by the evidence.
There's also a few things to say about conspiracies - for example, I think that at sufficiently large N values they scale worse than independence. Everyone telling the truth scales exponentially, as you said, but a conspiracy has an additional factor where everyone's story has to match with everyone else's, and everyone has to get along with everyone else - meaning, there's an N! factor against conspiracies. This is why we rightly consider any hypothesis involving "a vast conspiracy" to be a crackpot theory. Fortunately, we don't have to worry about this kind of calculation, given that we have the historical data.
Lastly, I'm in total agreement with you that we should follow Truth wherever it leads. So I'm very glad for this discussion.
Here's a summary of the issues you brought up, and where I stand on them:
I agree that numbers like 1e300, 1e100, or even 1e54 are too large as the final, overall probability values, as they're beyond the limit of how certain we can be. I'm also backing up from my final value of 1e32 as the probability for the resurrection, which I thankfully did not state too forcefully too often.
I still think 1e54 or 1e44 can be used as Bayes' factors, given that they are the factors between two specific hypothesis and not the factors for the whole "true" or "false" hypothesis in aggregate. They should not be thrown out just for being too large.
I still think that 1e8 is a good estimate for the Bayes' factor for a human testimony, and in fact our discussion here has only strengthened my belief. I can perhaps be talked down to 1e7, or 1e6 in set circumstances, but as a rough, order-of-magnitude estimate, 1e8 is a perfectly serviceable value. My confidence in this has actually been strengthened considerably as a result of our exchange.
I agree that the "license plate effect" is real, and it has a number of fascinating and important implications. But it does not really affect my calculated Bayes' factor of 1e54. Instead, it's main function is to increases the Bayes' factor for a human testimony when it applies. This is what allows us to do things like believing that the Gospels, as a whole, are reliable.
Many of the testimonies for the resurrection can be considered independent, until they successfully knock out all "reasonable" priors, leaving only things like conspiracy theories as the leftovers. I agree that at this point, the possibilities for strongly correlated testimonies must be considered.
I agree that the odds for the resurrection is still very high even after considering things like conspiracy theories.
Thank you again for taking the time to read my series and replying to me, Aron. Your reply was intellectually stimulating, and very useful!
(The series on the resurrection will continue on with this post.)
You may next want to read:
The principle of least awesomeness
The dialogue between two aliens who found a book on Earth
Another post, from the table of contents