Predictive Characterization of Mixtures of Markov Chains

TL;DR

This paper establishes necessary and sufficient predictive conditions to characterize Markov exchangeable processes, aiding Bayesian inference on Markov chains and providing minimal criteria for such stochastic models.

Contribution

It introduces new predictive criteria that precisely characterize Markov exchangeability and recurrence, advancing the theoretical understanding and practical application of Bayesian models for Markov processes.

Findings

Provides minimal sufficient predictive conditions for Markov exchangeability

Characterizes when predictive schemes define a prior for Markov chains

Illustrates applications with examples and new constructions

Abstract

Predictive constructions are a powerful way of characterizing the probability law of stochastic processes with certain forms of invariance, such as exchangeability or Markov exchangeability. When de Finetti-like representation theorems are available, the predictive characterization implicitly defines the prior distribution, starting from assumptions on the observables; moreover, it often helps designing efficient computational strategies. In this paper we give necessary and sufficient conditions on the sequence of predictive distributions such that they characterize a Markov exchangeable probability law for a discrete valued process X. Under recurrence, Markov exchangeable processes are mixtures of Markov chains. Thus, our results help checking when a predictive scheme characterizes a prior for Bayesian inference on the unknown transition matrix of a Markov chain. Our predictive conditions are in some sense minimal sufficient conditions for Markov exchangeability; we also provide predictive conditions for recurrence. We illustrate their application in relevant examples from the literature and in novel constructions.