A class of measure-valued Markov chains and Bayesian nonparametrics

TL;DR

This paper introduces a new class of measure-valued Markov chains based on exchangeable sequences, explores their asymptotic properties, and discusses applications in Bayesian nonparametric mixture modeling and chain generalizations.

Contribution

It proposes a novel class of measure-valued Markov chains, extending previous models, with detailed asymptotic analysis and applications in Bayesian nonparametrics.

Findings

New class of measure-valued Markov chains introduced

Asymptotic properties derived for the new class

Applications to Bayesian nonparametric mixture models discussed

Abstract

Measure-valued Markov chains have raised interest in Bayesian nonparametrics since the seminal paper by (Math. Proc. Cambridge Philos. Soc. 105 (1989) 579--585) where a Markov chain having the law of the Dirichlet process as unique invariant measure has been introduced. In the present paper, we propose and investigate a new class of measure-valued Markov chains defined via exchangeable sequences of random variables. Asymptotic properties for this new class are derived and applications related to Bayesian nonparametric mixture modeling, and to a generalization of the Markov chain proposed by (Math. Proc. Cambridge Philos. Soc. 105 (1989) 579--585), are discussed. These results and their applications highlight once again the interplay between Bayesian nonparametrics and the theory of measure-valued Markov chains.