Saturday, October 3, 2009


According to frequentism, the probability of an event E happening is equal to the limit of NE(n)/n as n goes to infinity, where N(n) is the number of times that E-type outcome occurs in the first n independent trials. (If there are only finitely many trials in the history of the universe, we've got a serious problem, since then we get the conclusion—surely inconsistent with current physics—that all probabilities are rational numbers. I am guessing that in that case we need to make a counterfactual move—if we were to go to infinity, what limit would we get?)

But now here is a puzzle for the frequentist: Why is it that N(n)/n in fact has a limit at all? The non-frequentist has an answer—the Law of Large Numbers implies that, with probability one, N(n)/n converges to the probability of E, if E has a probability. But it would be circular for the frequentist to offer this explanation.

1 comment:

Brandon said...

If I understand Van Mises-style frequentism correctly (I might not), I don't think the frequentist is committed to saying that N(n)/n must have a limit. If we ask why the frequentist proceeds as if it does, we would get two possible answers, depending on whether the question were a general one or about this particular case:

(1) If general, because it makes the model more usable;

(2) If particular, because it works for practical purposes, or is analogous to cases where it does so.

Since we're talking about hypothetical infinite frequentism here, probability is taken to be a feature of an idealized sequence that is being used to model the event. Thus the issue really boils down to a question about the application of idealized models to the real world: the ultimate desiderata for any model are (a) that it is more elegant/simple/convenient than alternatives; and (b) that it helps you get workable conclusions. So the question about why one would take N(n)/n to converge to a limit would be taken as a question about whether it makes the model simpler or easier to work within (which presumably it does) or whether it makes it easier to apply to the actual world (which presumably it does).