On approximation of smoothing probabilities for hidden Markov models

TL;DR

This paper demonstrates that smoothing probabilities in hidden Markov models can be effectively approximated using double-sided HMMs under general conditions, enabling ergodic theorems application and convergence of segmentation risks.

Contribution

It introduces a method to approximate HMM smoothing probabilities with double-sided models, facilitating analysis and convergence results.

Findings

Exponential forgetting property holds under general conditions.

Smoothing probabilities can be approximated by double-sided HMMs.

Pointwise MAP segmentation risks converge.

Abstract

We consider the smoothing probabilities of hidden Markov model (HMM). We show that under fairly general conditions for HMM, the exponential forgetting still holds, and the smoothing probabilities can be well approximated with the ones of double sided HMM. This makes it possible to use ergodic theorems. As an applications we consider the pointwise maximum a posteriori segmentation, and show that the corresponding risks converge.