Non Parametric Hidden Markov Models with Finite State Space: Posterior Concentration Rates

TL;DR

This paper provides theoretical guarantees for Bayesian nonparametric hidden Markov models with finite states, establishing posterior concentration rates for both discrete and continuous observations.

Contribution

It develops a general theorem for posterior concentration rates and applies it to derive minimax rates for models with Dirichlet process emissions.

Findings

Established posterior concentration rates in $L_1$-norm.

Derived minimax rates for discrete and continuous emission models.

Provided theoretical guarantees for Bayesian nonparametric HMMs.

Abstract

The use of non parametric hidden Markov models with finite state space is flourishing in practice while few theoretical guarantees are known in this framework. Here, we study asymptotic guarantees for these models in the Bayesian framework. We obtain posterior concentration rates with respect to the $L_1$-norm on joint marginal densities of consecutive observations in a general theorem. We apply this theorem to two cases and obtain minimax concentration rates. We consider discrete observations with emission distributions distributed from a Dirichlet process and continuous observations with emission distributions distributed from Dirichlet process mixtures of Gaussian distributions.