Efficient Bayesian estimation and use of cut posterior in semiparametric hidden Markov models

TL;DR

This paper develops efficient Bayesian methods for estimating transition matrices and emission densities in semiparametric hidden Markov models, introducing a cut posterior approach with proven contraction rates and practical simulations.

Contribution

It introduces a modular Bayesian estimation framework using cut posterior for HMMs, with new contraction rate results and practical implementation strategies.

Findings

Efficient Bayesian estimators for transition matrices are proposed.

A general theorem on contraction rates for the cut posterior approach is established.

Simulations demonstrate the theoretical results and ease of implementation.

Abstract

We consider the problem of estimation in Hidden Markov models with finite state space and nonparametric emission distributions. Efficient estimators for the transition matrix are exhibited, and a semiparametric Bernstein-von Mises result is deduced. Following from this, we propose a modular approach using the cut posterior to jointly estimate the transition matrix and the emission densities. We derive a general theorem on contraction rates for this approach. We then show how this result may be applied to obtain a contraction rate result for the emission densities in our setting; a key intermediate step is an inversion inequality relating $L^1$ distance between the marginal densities to $L^1$ distance between the emissions. Finally, a contraction result for the smoothing probabilities is shown, which avoids the common approach of sample splitting. Simulations are provided which demonstrate both the theory and the ease of its implementation.