Bayesian definition of random sequences with respect to conditional probabilities

We study Martin-Lof random (ML-random) points on computable probability measures on sample and parameter spaces (Bayes models). We consider variants of conditional randomness defined by ML-randomness on Bayes models and those of conditional blind randomness. We show that variants of conditional blind randomness are ill-defined from the Bayes statistical point of view. We prove that if the sets of random sequences of uniformly computable parametric models are pairwise disjoint then there is a consistent estimator for the model. Finally, we present an algorithmic solution to a classical problem in Bayes statistics, i.e. the posterior distributions converge weakly to almost all parameters if and only if the posterior distributions converge weakly to all ML-random parameters.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article examines the variants of conditional randomness and conditional blind randomness defined by ML-randomness on Bayes models. It is shown that variants of conditional blind randomness are ill-defined from the Bayes statistical point of view. The paper also proves the existence of a consistent estimator for the model when the sets of random sequences of uniformly computable parametric models are pairwise disjoint. Furthermore, an algorithmic solution to a classical problem in Bayes statistics is presented, indicating that the posterior distributions converge weakly to almost all parameters if and only if the posterior distributions converge weakly to all ML-random parameters. (c) 2023 Elsevier Inc. All rights reserved.