P(H/D) = [P(D/H) x P(H)] ÷ P(D).
That is one common formulation of Bayes' theorem, and at the heart of the view of statistics associated with a mid-18th century Englishman, Thomas Bayes.
I blogged about this in January. If you don't understand the notation above, and would like a primer, go here.
I bring it up again because Bayesianism has attracted new attention in the blogosphere.
Sometimes the above is called "Bayes' rule," because it suggests a rule for constantly updating your views of the likelihood of events. Your hypothesis of probability of H on Tuesday (the prior) is to be updated according to what happens or doesn't happen Wednesday, thereby yielding the posterior probability.
Econoblogger Noah Smith wrote a post he calls "Bayesian Superman." He combines a (male) teenager's sense of invincibility with something like Pascal's wager, and proposes the following thought experiment:
Suppose that you believe that there is a nonzero probability that H is true, where H in this case is the hypothesis, "God is watching out for me, has a special purpose in mind for me, and so whatever the apparent dangers, He will not allow my death prior to the accomplishment of that purpose."
What Smith is suggesting first is that H may have been adopted however you like, perhaps with a wagering element to it. After all, although today's working hypothesis is a modification of yesterday's prior, and yesterday's prior was at some point a posterior after the modification of the previous day's prior ... acknowledging all that, this cannot be an infinite string in any particular life. There was a first H, the prior of priors.
So we get to the tricky part. You are a faithful Bayesian, updating your hypotheses day by day. You did not die yesterday. You did not die the day before. And so forth. If you have entertained H as a hypothesis for many days, then presumably your confidence in H has increased with each sunset, where the relevant datum, D, is simply "I didn't die today".
So, we get somewhat further than Pascal would allow here. Not only might Pascal argue that the original wager H is rational, but Bayes would butt in to say that this ever increasing confidence in it is likewise rational.
Noah Smith wrote a think piece -- some of the above is a rather loose paraphrasing of it -- in order to explore the "observed behavior of teenagers," who seem to be Bayesians by instinct. Another blogger, Lars P. Syll, has picked up on the thought experiment and usef it as an attack on Bayesianism, which he describes as an internally consistent system of thought that nonetheless threatens harm to science.
Smith, on his twitter account, says that he didn't really intend it the way Syll took it, but he does think Syll makes a good point.
There is a lot to think about here. One obvious problem with Smith's thought experiment, though, is survivorship bias. Starting with secularist premises if I may: only those who have not yet died can be said to draw inferences at all, Bayesian or otherwise. Someone who has died doesn't then have the luxury of saying, "aha! there is now a zero probability of H."
Smith notices the survivorship issue, then waves his hands a bit around it.
I suspect (a related point) that Bayes would have had a difficulty with premises that need the first-person pronoun "I" or "me" to make any sense. Certainly lemming C can notice that his friends, lemmings A and B, both died after jumping off yonder cliff, and that would decrease to a rational lemming the plausibility of any hypothesis that makes the act of jumping off a cliff seem a safe one. The fact that A and B can't be aware of this inference shouldn't keep C from drawing it.
Whatever one's take on God, Ubermensch, or teenagers, I don't think that Smith's meditation makes much of a dent in the case for Bayesian statistics, under any non-parodic understanding of the latter.