Thursday, July 3, 2008

The Documentary Hypothesis, Torah Min Hashamayim, and Bayes' Theorem - Three More Implications

Following up on my earlier post, there are at least three more implications about the use of Bayes' Theorem, after the 5 listed there.

6. An additional comment about prior probabilities and "assuming away the problem." There is nothing wrong with prior probabilities; we use them all the time. Here's a simple example.

Suppose your good friend was supposed to meet you at 7:00 to see a movie that starts at 7:20. It's 7:15 and he has not shown up. You assume he is late, but then ha-Satan shows up and makes a series of clever argument (isn't that just like him): Your friend is not late, but he really hates you. He is ending your friendship and deliberately making you wait solely to inconvenience you.

This is certainly possible, but your prior probability estimate of that would be quite low. After all, he is your good friend. But now ha-Satan starts piling on the evidence. Your friend has a cellphone; he would certainly call if he were running late. Hmmm. Good point. Your friend has always been very reliable, and thus the statistical odds of him running 15 minutes late when you only have 20 minutes to spare are very small. Hmmm. Good point. Your friend really wanted to see the movie, and in fact you two spoke about it in the late afternoon and specifically discussed both the time and place. He is not likely to have forgotten or gone to another theater. Hmmm. Good point. And so it goes.

Each of these facts is easily explained by the friend-hates-you theory, but not so easily explained by the friend-is-just-late theory. Using Bayes' Theorem, these facts are shifting the probability towards the friend-hates-you theory. But if your estimated prior probability of this theory is microscopically small (say, 1 in a billion --- I'll plug in numbers here just to illustrate), then you might estimate the odds now at 1 in 1 million. You now conclude that it is 1000 times more likely that your friend really hates you, but you still place the odds at something so microscopically small that it is still essentially zero.

That's the way some TMH / DH debates go. (More often on the pro-DH side, but not always.) One person keeps making good points, and the other person keeps offering weak responses but is just not budging. The proponent gets agitated, the opponent seems unfazed, and each side thinks the other is crazy. I think what is happening in those debates is the opponent simply assigns such a high prior probability to his theory, that the net effect of the proponent's arguments is simply to move the probability of that theory from almost zero to a slightly larger almost zero.

7. There's another unresolved issue: what should the prior probability be? There's no good way to answer that question since it depends to one's subjective beliefs before any evidence. For whatever reason, some of us tend to think that the
idea that God wrote the Torah is obviously true and others of us tend to think that the idea that people wrote the Torah is obviously true.

I think each person must decide this issue for his or her self. It might be useful as a heuristic device simply to assume the prior probability is 50%. That is, assume that each theory is equally likely and then go look at the evidence. The advantage of this approach is that avoids biasing the outcome based on initial probabilities. The disadvantage is that no one really estimates this at 50%. All of our prior intuitions is that TMS is either likely or not likely, but not exactly equal to the not-TMH. But this might all be academic. Pick a probability for the sake of the discussion, and one can always revise it later.

8. The evidence can persuade even a harsh skeptic. Go back to the jar examples, and take two really ambiguous jars with the following percentage of yellow, green, blue, and red marbles:

Type 1: 40%, 25%, 20%, 15%
Type 2: 30%, 20%, 15%, 35%

If you draw (say) a yellow marble out, it is just not going to shift the odds that much, regardless of your prior probabilities. Type 1 jars have 40% yellow marbles and Type 2 jars have 30%. So a yellow marble just does not give you much information.

But suppose you randomly draw 1000 marbles (with replacement, for you math geeks out there). And you find that your random sample gives the following probabilities

Sample: 39%, 26%, 21%, 14%

The probability of drawing this from a Type 1 jar is quite high, and the probability of drawing this from a Type 2 jar is quite low. (Someone can crunch the numbers if they can find a standard normal chart with that many standard deviations.) So even if your prior odds were that there was only a 1 in a million chance that you had a Type 1 jar, the evidence here is so compelling that it would be a virtual certainty that you did in fact have a Type 1 jar.

The implication of this is that regardless of whether one initially thinks TMH is highly likely or the DH is highly likely, enough evidence consistent with the other theory and inconsistent with your should persuade you. And this is true even if the evidence is somewhat ambiguous, so long as it fits better with the other theory.

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