About the posterior distribution in hidden Markov models with unknown number of states

TL;DR

This paper analyzes Bayesian methods for finite state hidden Markov models with unknown states, providing asymptotic results for posterior concentration and a consistent estimator for the number of states.

Contribution

It offers the first frequentist asymptotic evaluation of Bayesian procedures in HMMs with unknown states, including posterior concentration rates and a method to estimate the number of hidden states.

Findings

Posterior concentrates at optimal rates for marginal densities.

A Bayesian estimator for the number of states is consistent.

Conditions on the prior prevent nonstandard likelihood ratio behaviors.

Abstract

We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that is for the density of a fixed number of consecutive observations. Using conditions on the prior, we are then able to define a consistent Bayesian estimator of the number of hidden states. It is known that the likelihood ratio test statistic for overfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. Inference of parameters is a much more difficult task than inference of marginal densities, we still provide a precise description of the situation when the observations are i.i.d. and we allow for $2$ possible hidden states.