Skip to main content

I'm safe: I'm on a mission from God











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.







Comments

Post a Comment

Popular posts from this blog

A Story About Coleridge

This is a quote from a memoir by Dorothy Wordsworth, reflecting on a trip she took with two famous poets, her brother, William Wordsworth, and their similarly gifted companion, Samuel Taylor Coleridge.



We sat upon a bench, placed for the sake of one of these views, whence we looked down upon the waterfall, and over the open country ... A lady and gentleman, more expeditious tourists than ourselves, came to the spot; they left us at the seat, and we found them again at another station above the Falls. Coleridge, who is always good-natured enough to enter into conversation with anybody whom he meets in his way, began to talk with the gentleman, who observed that it was a majestic waterfall. Coleridge was delighted with the accuracy of the epithet, particularly as he had been settling in his own mind the precise meaning of the words grand, majestic, sublime, etc., and had discussed the subject with William at some length the day before. “Yes, sir,” says Coleridge, “it is a majestic wate…

Hume's Cutlery

David Hume is renowned for two pieces of cutlery, the guillotine and the fork.

Hume's guillotine is the sharp cut he makes between "is" statements and "ought" statements, to make the point that the former never ground the latter.

His "fork" is the division between what later came to be called "analytic" and "synthetic" statements, with the ominous observation that any books containing statements that cannot be assigned to one or the other prong should be burnt.

Actually, I should acknowledge that there is some dispute as to how well or poorly the dichotomy Hume outlines really maps onto the analytic/synthetic dichotomy. Some writers maintain that Hume meant something quite different and has been hijacked. Personally, I've never seen the alleged difference however hard they've worked to point it out to me.

The guillotine makes for a more dramatic graphic than a mere fork, hence the bit of clip art above.

I'm curious whe…