Metropolising forward particle filtering backward sampling and Rao-Blackwellisation of Metropolised particle smoothers

Smoothing in state-space models amounts to computing the conditional distribution of the latent state trajectory, given observations, or expectations of functionals of the state trajectory with respect to this distributions. For models that are not linear Gaussian or possess finite state space, smoothing distributions are in general infeasible to compute as they involve intergrals over a space of dimensionality at least equal to the number of observations. Recent years have seen an increased interest in Monte Carlo-based methods for smoothing, often involving particle filters. One such method is to approximate filter distributions with a particle filter, and then to simulate backwards on the trellis of particles using a backward kernel. We show that by supplementing this procedure with a Metropolis-Hastings step deciding whether to accept a proposed trajectory or not, one obtains a Markov chain Monte Carlo scheme whose stationary distribution is the exact smoothing distribution. We also show that in this procedure, backward sampling can be replaced by backward smoothing, which effectively means averaging over all possible trajectories. In an example we compare these approaches to a similar one recently proposed by Andrieu, Doucet and Holenstein, and show that the new methods can be more efficient in terms of precision (inverse variance) per computation time.