  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.
