## Sunday, April 25, 2010

### Another argument against frequentism

Suppose that the probability of a process Q being followed by A is given by the ratio of the number of times it is followed by A to the number of times the process occurs. Suppose that Q is run only finitely many times in the history of the universe, and that the frequency is some number f(A,Q) strictly between 0 and 1. Here is an argument against the claim that f(A,Q) is equal to the probability P(A|Q) that Q is followed by A: if the number n of occurrences of Q is large, and the occurrences are independent, the probability distribution of the observed frequency of As will be approximately a Gaussian centered on P(A|Q) with a standard deviation proportional to n−1/2. Because a Gaussian is flat around its peak, the probability that the observed frequency of As will be exactly equal to P(A|Q) is small (and tends to zero as n tends to infinity). Hence, it is unlikely that the observed frequency for an independent process that occurs many times should equal the probability. But if frequentism is true, it is certain that the observed frequency equals the probability. The only way this could be true, assuming the frequency is strictly between 0 or 1, is if the process is not independent. Therefore, if frequentism is true for a process with a frequency strictly between 0 and 1, the process is not an independent one. But sure it is possible to have independent processes with frequencies strictly between 0 and 1. Hence, frequentism is not true.