Perfect sampling for nonhomogeneous Markov chains and hidden Markov models

TL;DR

This paper introduces a perfect sampling method for nonhomogeneous Markov chains and hidden Markov models, enabling exact sampling of the signal process conditioned on observations with finite data, based on ergodicity and coupling concepts.

Contribution

It provides a novel perfect sampling characterization for nonhomogeneous Markov chains and applies it to hidden Markov models, introducing an efficient algorithm requiring finite observations.

Findings

Characterization of weak ergodicity via perfect sampling

Development of an algorithm for exact sampling from hidden Markov models

Link between successful coupling, ergodicity, and filter stability

Abstract

We obtain a perfect sampling characterization of weak ergodicity for backward products of finite stochastic matrices, and equivalently, simultaneous tail triviality of the corresponding nonhomogeneous Markov chains. Applying these ideas to hidden Markov models, we show how to sample exactly from the finite-dimensional conditional distributions of the signal process given infinitely many observations, using an algorithm which requires only an almost surely finite number of observations to actually be accessed. A notion of "successful" coupling is introduced and its occurrence is characterized in terms of conditional ergodicity properties of the hidden Markov model and related to the stability of nonlinear filters.