Monday, February 22, 2016

Frequentism and explanation

This is really very obvious, and no doubt in the literature, but somehow hasn't occurred to me until now. Suppose that a fair coin is tossed an infinite number times. Suppose, further, than in the first hundred tosses it lands heads about half the time. It's no mystery why it lands heads about half the time in the first hundred tosses: it's because the probability of heads is 1/2 (plus properties of the binomial distribution). But suppose frequentism is true. Then the reason the probability of heads is 1/2 is that the infinite sequence has a limiting proportion of heads of 1/2. Now consider these three statements:

  • A: approximately half of the first 100 tosses are heads
  • B: the limiting proportion of heads is 1/2
  • C: the limiting proportion of heads starting with the 101st toss is 1/2.
Then, C is statistically independent of A, as A depends on the first 100 tosses and C depends on the other tosses. Clearly, C has no explanatory power with respect to A. But B is logically equivalent to C (the first 100 tosses disappear in the limit). How can B explain A, then?

There are some gaps in the argument--explanation is hyperintensional, for instance. But I think the argument has a lot of intuitive force.


Heath White said...

I'm not sure, but... I would think that on frequentism, probability claims do not have any explanatory force. To say a coin has a probability of 1/2 of landing heads, is just to say that in [some large population of coin tosses] it lands heads 1/2 the time. But the probability claim is explained by the way the coin lands, not vice versa.

Similarly: if natural laws are just Humean regularities, then natural laws describe the universe pretty well but they don't explain it, in any normal sense.

Alexander R Pruss said...

Humean laws are supposed to explain in some weird global way.