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P.A. Sturrock (Stanford University)
In problems of the Bernoulli type, an experiment or observation yields a count of the number of occurrences of an event, and this count is compared with what it to be expected on the basis of a specified and unremarkable hypothesis. The goal is to determine whether the results support the specified hypothesis, or whether they indicate that some extraordinary process is at work. This evaluation is often based on the ``p-value" test according to which one calculates, on the basis of the specific hypothesis, the probability of obtaining the actual result or a ``more extreme" result. Textbooks caution that the p-value does not give the probability that the specific hypothesis is true, and one recent textbook asserts ``Although that might be a more interesting question to answer, there is no way to answer it."
The Bayesian approach does make it possible to answer this question. As in any Bayesian analysis, it requires that we consider not just one hypothesis but a complete set of hypotheses. This may be achieved very simply by supplementing the specific hypothesis with the maximum-entropy hypothesis that covers all other possibilities in a way that is maximally non-committal. This procedure yields an estimate of the probability that the specific hypothesis is true. This estimate is found to be more conservative than that which one might infer from the p-value test.