Bayesian Inference Tempered by Classical Hypothesis Testing ∗

When a 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 this practice (under suitable background assumptions) will cause dynamic inconsistency. He may wish to limit this inconsistency. We show that if these non-Bayesian updates, which we call “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 consistent. 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 results give the decision-maker some latitude to violate Bayes’ rule while bounding the resulting inconsistency, and justifies the classical criterion for hypothesis testing on decision-theoretic grounds. ∗Previous versions were circulated as “Provisional Beliefs and Paradigm Shifts” and “A Bayesian Foundation for Classical Hypothesis Testing” †I am grateful for conversations with Nabil Al-Najjar, Eddie Dekel, Tai-Wei Hu, Ehud Kalai, Eric Maskin, Allegra Petti, Phil Reny, Larry Samuelson, and Muhamet Yildiz, and seminar participants at Northwestern University, the Transatlantic Theory Conference, University of Montreal, the Stony Brook Game Theory Festival, and University of Pennsylvania. “...I equate the rational attitude and the critical attitude. The point is that, whenever we propose a solution to a problem, we ought to try as hard as we can to overthrow our solution, rather than defend it. Few of us, unfortunately, practice this precept...” Karl Popper, The Logic of Scientific Discovery