Posterior consistency for partially observed Markov models

TL;DR

This paper proves that the Bayesian posterior distribution converges to the true parameter in partially observed Markov models, under mild conditions, extending previous results to non-compact spaces and non-stationary data.

Contribution

It establishes posterior consistency for a broad class of partially observed Markov models, including non-compact and non-stationary cases, linking it to maximum likelihood estimator consistency.

Findings

Posterior consistency holds under mild assumptions.

Results extend to non-compact state and parameter spaces.

Applications include Gaussian linear models and stochastic volatility models.

Abstract

In this work we establish the posterior consistency for a parametrized family of partially observed, fully dominated Markov models. As a main assumption, we suppose that the prior distribution assigns positive probability to all neighborhoods of the true parameter, for a distance induced by the expected Kullback-Leibler divergence between the family members' Markov transition densities. This assumption is easily checked in general. In addition, under some additional, mild assumptions we show that the posterior consistency is implied by the consistency of the maximum likelihood estimator. The latter has recently been established also for models with non-compact state space. The result is then extended to possibly non-compact parameter spaces and non-stationary observations. Finally, we check our assumptions on examples including the partially observed Gaussian linear model with correlated noise and a widely used stochastic volatility model.