Bayesian definition of random sequences with respect to conditional probabilities

We study Martin-L\"{o}f random (ML-random) points on computable probability measures on sample and parameter spaces (Bayes models). We consider four variants of conditional random sequences with respect to the conditional distributions: two of them are defined by ML-randomness on Bayes models and the others are defined by blind tests for conditional distributions. We consider a weak criterion for conditional ML-randomness and show that only variants of ML-randomness on Bayes models satisfy the criterion. We show that these four variants of conditional randomness are identical when the conditional probability measure is computable and the posterior distribution converges weakly to almost all parameters. We compare ML-randomness on Bayes models with randomness for uniformly computable parametric models. It is known that two computable probability measures are orthogonal if and only if their ML-random sets are disjoint. We extend these results for uniformly computable parametric models. 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.Comment: revised versio