Monday, December 10, 2007

A Problem for Frequentists?

I don’t know much about the philosophy of probability yet, but I wanted to throw an objection to the frequentist interpretation out there and see whether it sticks. Consider a pair of propositions p and q. From the dawn of time until 10,000 C.E., p is true 99 out of every 100 times q is true. Subsequently, p is true 1 out of every 100 times q is true. Suppose that q turns out to be true only some finite number of times into the future, but on many occasions for many trillions of years after 10,000 C.E. Let m be the number of times q turns out to be true in the whole history of the universe before 10,000 C.E., and n be the number of times q is true after 10,000 C.E. On the frequentist interpretation, I think P(p | q) = .99*[m / (m + n)] + .01*[n / (m + n)]. This number will, evidently, be quite lower than .99; if n is great enough, the frequentist’s value for P(p | q) will approach .01. But wouldn’t it be extremely counterintuitive, long before 10,000 C.E., to say that P(p | q) is far below .99, even approaching .01? After all, p follows on q nearly all the time and will for another 7,993 or so years. So it seems that the frequentist interpretation yields the wrong results in this case.

Perhaps my intuition here is that statements like this about the probability of untensed propositions are themselves tensed. If this intuition is commonly shared, then a frequentist interpretation of "P(p | q)" should refer to something like the number of times at which p is true divided by the number of times q is true for some length of time before and after the present moment. But then the frequentist needs to say something about why this length of time should be of any particular size. I have some ideas about what she can say, but I'm having trouble expressing them precisely, so I'll save them for another post.

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