I want to try to understand Kyler's work in light of some of Billy's observations.
Bayes forces you to account for your misses as well as your hits. My impression is that apologists find their way to make Bayes do the opposite, and amplify their hits instead.
The absolute best explanation of Bayes that exists on the Internet is
when there's smoke, there's fire on mathisfun.com.
Notice in the example, P(B) is the probability of seeing smoke on its own. That's the crucial component that accounts for your misses. In the example, dangerous fires are rare, they nearly always produce smoke, but because smoke is relatively common, at 10% on its own (bbq, fireworks), the probability of a dangerous fire when you see smoke is only 9%. If the probability of smoke on its own were 90%, then seeing smoke would tell us nothing we didn't already know.
Crucial to our understanding of P(B) is that it covers
all possibilities of smoke. If we select only certain circumstances by which we might view the existence of smoke, we will underrepresent the commonality of smoke. In general this is what I think apologists do; what Kyler is doing. And I think I understand his justification for it.
P(A) = probability that prayer healed me. P(B) = probability I healed under all circumstances. P (B|A) = probability I healed if prayed for
P (A|B) = P(A) * P(B|A) / P(B)
The probability prayer healed me given that I healed is inversely proportional to the probability I healed under all circumstances. Even if the probability I healed when prayed for isn't very good, say, 25%, if the probability I healed under all circumstances is very tiny, then prayer wins.
Allow me to recast Kyler's episodes into the simple form of Bayes, to be like the smoke-fire example.
numerator = God
-------
denominator = Nature
apologist goal: Make natural causes very tiny, and that inflates God, even if God isn't very good.
Episode 1.
P(A) = probability that Book of Mormon is ancient. P(B) = probability that the Book of Mormon is long (under all circumstances). P (B|A) = probability the Book of Mormon is long if it is ancient.
P(A|B) = the probability the book is ancient if it is long. Notice Kyler only gave P(B|A) 50-50 odds. All the work is done by the tiny probability that the Book of Mormon can be long under all circumstances.
Episode 3.
P(A) = probability God dictated. P(B) = probability Book of Mormon was produced quickly. P(B|A) = probability produced quickly if God dictated.
P(A|B) = probability God dictated given produced quickly. Billy rightly points out that P(B|A) = 1 is nonsense. If I could quibble just a little with Billy, I'd just point out that even if we substantially lower P(B|A), Kyler has made P(B) so utterly miniscule -- the probability of seeing smoke on it own -- that we can substantially lower the 1 to 1% or way lower than that and God still wins by a landslide.
How is Kyler constraining the probability of seeing smoke on its own, or the probability of Book of Mormon produced quickly in general or probability of Book of Mormon long under all circumstances as so utterly tiny? Hold that thought. Let's tackle episode 2, because Billy gave that one huge props.
Episode 2.
Suddenly, Kyler is a stats guru per Billy, he does an awesome analysis, at least when isolated from the broader project. How is he suddenly firing on all cylinders?
Billy wrote:the pattern of inconsistencies in the first vision accounts are unlikely if the event really happened, but those same inconsistencies are also unlikely if the whole thing was made up.
P(A) = probability story fabricated. P(B) probability of (these) inconsistencies in stories in general P(B|A) = probability of (these) inconsistencies if fabricated
P(A|B) = probability story was fabricated, given the inconsistencies. Aha!
numerator = Satan
---------
denominator = Nature
If the numerator is the case that Smith did something fraudulent given the evidence, then we definitely don't want to restrict smoke in general, or in this case, inconsistencies in general to a small subset of all the known ways that inconsistencies can happen.
Kyler is an intelligent guy and he's a straight-shooter, and I'm not saying this is bad faith or stupidity on his part.
Kyler wrote:you didn't have much to say on Episode 3 other than to present an alternate theory (a fair thing to do, though I don't think it works well in your specific case.
This line brought back a suspicion I had from the beginning. I think the more complex form of Bayes that pits hypothesis 1 against hypothesis 2 is feeding into a natural confirmation bias. Specifically, in the cases where hypothesis 1 is God (fire), hypothesis 2 is some extraordinarily knee-capped circumstance by which we'd find smoke, which doesn't represent the best way to find smoke, or the full range of smoke possibilities. In principle, there is nothing wrong with this; if you want to compare a semi-lame hypothesis to a total crap hypothesis, then your result might be correct that the lame hypothesis is far more likely. But that doesn't help when looking for the best hypothesis. As in the idea of opportunity cost, hyp 2 needs to be the best forgone alternative -- the most likely explanation you can think of aside from the idea you want to win.
My hunch is if Kyler were to go back to the drawing board and work through several examples of Bayesian reasoning in the simple form, rather than in hypothesis testing form, in order to force P(B) to be conceptually, the probability of healing to be healing on its own, or smoke as smoke on its own, then that might help with the intuition to see the strawman in his hyp 2s. I think it would help with the consistency problems Billy has complained about.