Frequentism is the second major branch of probability theory. It uses long-term frequency as the fundamental definition of probability. This does not require money to define. Unfortunately, frequentism can't tell us the probabilities we want to know, like the probability that if I take a certain drug it will help me, or the probability that I will make money buying a certain stock. It can only tell us about probabilities created by the experimenter, and not even about specific probabilities, just average probabilities of groups of predictions. In a frequentist interpretation of a drug trial, there is no estimate of the probability that the drug works, only of the probability that the randomization scheme for assigning subjects to treatment or control groups—randomness the experimenter created—produced the observed result under the assumption the drug had no effect. Things are actually worse for observational studies where the researcher does not create randomness, such as an econometric study of the effect of monetary policy on inflation. For these, the researcher makes a statement about the probability of randomness she pretends she created.

A frequentist might test hypotheses at the 5 percent level. She can tell us that in the long run, fewer than 5 percent of the hypotheses she rejects will turn out to be true. That's mathematically true (at least if her other assumptions are correct) without reference to a numeraire. But why would we care? What if the 95 percent ...

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