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

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.