Tuesday, January 31, 2006

Bayes Bogus?

Mr. Spock, of the original Star Trek, was pretty much the perfect straight man. He never got the joke, and he never got the girl. Which is why, I suppose, that Jim kept him on despite his limitations as a first officer. I am thinking here of his complete incompetence at elementary probability calculations. Whenever some dangerous mission was contemplated, Spock would portentiously announce some six or eight digit probability of failure, with success being somewhere out there in tens or hundred thousanths of a percent. Undeterred, Kirk would set out, usually taking with him the pilot, the ship's doctor, and the only key to the officers' bathroom. Naturally Spock was always wrong, but nobody ever complained or sent him back to remedial statistics.

I'm sure he must have somehow become confused about Bayesian inference. Using Bayes theorem to compute probabilities is of course perfectly legitemate and absolutely necessary for most statistical reasoning, but the Bayesian interpretation of probability drives me nuts. From the Wikipedia:

There are two broad categories of probability interpretations: Frequentists talk about probabilities only when dealing with well defined random experiments. The relative frequency of occurrence of an experiment's outcome, when repeating the experiment, is a measure of the probability of that random event. Bayesians, on the other hand, assign probabilities to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility.
Subjective plausibility makes my skin crawl. I have no idea what the hell it is supposed to mean. I can't think of any case when an expressed probability isn't either based on an explicit or implicit frequentist interpretation or else, like Mr. Spock's invariably misguided predictions, merely an ignorant prejudice.

Consider James Annan's
discussion of Bayesian probabilities referenced earlier. He talks about the "probability" that the number "1,234,567,897" is prime. Of course at this point I know it's zero, since it's divisible by 994817 as well as two two digit primes, but he discussed some general methods for guessing the probability of primality for an arbitrary 10 digit integer based on things like the asymptotic density of primes (number of primes less than x is asymptotic to x/(ln x)), as well as the simple tests for divisibility by 2, 3, and 5. Each of these estimates is based on a frequency interpretation. I don't see any room for any interpretation that isn't essentially frequentist.

The other essential element is theory. Without some theory connecting previously observed events with future events, no prediction is possible.