Schnorr Randomness and Effective Bayesian Consistency and Inconsistency

We study Doob’s Consistency Theorem and Freedman’s Inconsistency Theorem from the vantage point of computable probability and algorithmic randomness. We show that the Schnorr random elements of the parameter space are computably consistent, when there is a map from the sample space to the parameter space satisfying many of the same properties as limiting relative frequencies. We show that the generic inconsistency in Freedman’s Theorem is effectively comeager, which implies the existence of computable parameters which are not computably consistent. Taken together, this work provides a computability-theoretic solution to Diaconis and Freedman’s problem of “know[ing] for which [parameters] θ θ theta the rule [Bayes’ rule] is consistent” [4, p. 4], and it strengthens recent similar results of Takahashi [31] on Martin-Löf randomness in Cantor space.