A Bayesian Foundation for Classical Hypothesis Testing
Jonathan Weinstein · 2013
A decision-maker can ensure dynamic consistency by following Bayes ’ rule, but he may wish to balance such consistency against other goals. That is, when the decision-maker is surprised by a pattern unaccounted for in his prior, he may wish to change his beliefs in a way which violates Bayes’ rule, but he may also wish to limit his inconsistency. We show that if such non-Bayesian events, or “paradigm shifts, ” are rare, in the sense that they occur only with a small probability α according to the decision-maker’s initial belief, the decision-maker will be “approximately ” dynamically con-sistent. Our notion of “approximate ” dynamic consistency is that the possible arbitrage against the decision-maker is small compared to the size of his transactions. The quantity α is equivalent to the level of a classical hypothesis test, so our results provide a decision-theoretic foundation for the classical criteria for rejecting a null hypothesis. Our findings give the decision-maker some latitude to revise his model while bounding the pain of inconsistency, and unify the classical and Bayesian modes of inference. ∗A previous version was circulated as “Provisional Beliefs and Paradigm Shifts.”