# Bayesian definition of random sequences with respect to conditional probabilities

## TL;DR

This paper explores the concept of ML-randomness in Bayesian models, analyzing conditional randomness variants, their well-definedness, and the relationship between posterior convergence and ML-random parameters.

## Contribution

It introduces a new perspective on conditional randomness in Bayesian models and provides an algorithmic solution to posterior convergence issues related to ML-random parameters.

## Key findings

- Conditional blind randomness variants are ill-defined in Bayesian context.
- Existence of a consistent estimator when model sets are pairwise disjoint.
- Posterior distributions converge to ML-random parameters if and only if they converge weakly to all such parameters.

## Abstract

We study Martin-L\"{o}f 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.