Marginal Likelihood Computation for Hidden Markov Models via Generalized Two-Filter Smoothing

TL;DR

This paper introduces an unbiased Monte Carlo estimate for the marginal likelihood in hidden Markov models using a generalized two-filter smoothing approach, with theoretical guarantees and numerical analysis.

Contribution

It presents the first unbiased Monte Carlo estimator for the marginal likelihood in HMMs based on the generalized two-filter smoothing, with proven CLT and theoretical analysis.

Findings

Estimator is unbiased and satisfies a CLT.

Theoretical results extend to expectation estimates w.r.t. marginal distributions.

Numerical experiments demonstrate the estimator's practical effectiveness.

Abstract

In this note we introduce an estimate for the marginal likelihood associated to hidden Markov models (HMMs) using sequential Monte Carlo (SMC) approximations of the generalized two-filter smoothing decomposition (Briers, 2010). This estimate is shown to be unbiased and a central limit theorem (CLT) is established. This latter CLT also allows one to prove a CLT associated to estimates of expectations w.r.t. a marginal of the joint smoothing distribution; these form some of the first theoretical results associated to the SMC approximation of the generalized two-filter smoothing decomposition. The new estimate and its application is investigated from a numerical perspective.