Schnorr Randomness and Effective Bayesian Consistency and Inconsistency

TL;DR

This paper explores the relationship between computable probability, algorithmic randomness, and Bayesian consistency, demonstrating that Schnorr random parameters are computably consistent and analyzing the effective genericity of Freedman's inconsistency.

Contribution

It provides a computability-theoretic framework for understanding Bayesian consistency and inconsistency, extending results on Schnorr and Martin-Löf randomness in this context.

Findings

Schnorr random elements of the parameter space are computably consistent.

Freedman's generic inconsistency is effectively generic, leading to non-computable consistent parameters.

The work offers a computability-based solution to Bayesian consistency problems.

Abstract

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 generic, 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] the rule [Bayes' rule] is consistent'', and it strengthens recent similar results of Takahashi on Martin-L\"of randomness in Cantor space.