Wednesday, March 24, 2010

Why frequentism about probabilities is wrong

It would be surprising if the following argument were not already known. But I rather like the argument, so I'll post it. According to frequentism as I shall use the term, the probabilities of a random event are to be read off from actual infinite-run frequencies. Here is a disproof of frequentism.

  1. It is logically possible to have a physically-realized random variable (a) whose probability measure is atom-free, i.e., has the probability that the measure of a singleton set is zero—uniform and Gaussian measures on continua have this property, and (b) the random variable is physically realized on a countable infinity of independent occasions in the history of the universe.
  2. If frequentism is true, (1) is false.
  3. Therefore, frequentism is not true.

Argument for (1): Scientists all the time model phenomena with atom-free measures. Consider, for instance, decay times of radioactive substances, or various Gaussian models. These models may not be entirely accurate, but it would be an amazing result if these models for basic philosophical reasons could not be right. (Besides, and this is more of an ad hominem, frequentism tends to be liked by folks of a reductionistic sort, and these folks tend to be in awe of science and unwilling to dictate to scientists what theories are and are not acceptable.)

Argument for (2): Suppose that in some world w we have physically realized a countable infinity of independent copies, X1, X2, ..., of an atom-free random variable X (a random variable whose associated measure is atom-free, i.e., a random variable such that the probability of its having any particular value is always zero). There are two versions of frequentism that I shall address: on one, we look at actual frequencies in an infinite run, and on the other, we consider the subjunctive conditional were the experiment repeated countably infinitely many times, what would the frequencies be? In w, however, the two come to the same thing in respect of X, because in w the experiment already is repeated countably infinitely many times. Let x1,x2,... be the outcomes of X1,X2,.... Let B be any Borel-measurable set in the space where X takes its values (e.g., a Borel susbet of the reals). Then,

  1. If frequentism of either variety is true, then in w we have: P(X is in B) = lim N(B,n)/n
where the limit is taken as n goes to infinity, and N(B,n) is the number of values of x1,...,xn that are in the set of B. Now, let B be the set {x1,x2,...}. Then:
  1. lim N(B,n)/n = 1
since all of the xi are in B. Therefore:
  1. If frequentism of either variety is true, then in w we have: P(X is in B)=1.
But the following is a theorem:
  1. If B is any countable set and X is a variable whose distribution is atom-free, then P(X is in B)=0.
(This follows from countable additivity and being atom-free—just break B up into countably many singletons, note that each singleton has zero measure, and hence so does their union.) Therefore, if frequentism is true, the scenario that defined w cannot be possible.

1 comment:

Alexander R Pruss said...

This is one of Hajek's 15 arguments against frequentism. One can get out of this problem in this way.