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Smoothing in state-space models amounts to computing the conditionaldistribution of the latent state trajectory, given observations, orexpectations of functionals of the state trajectory with respect to thisdistributions. For models that are not linear Gaussian or possess finite statespace, smoothing distributions are in general infeasible to compute as theyinvolve intergrals over a space of dimensionality at least equal to the numberof observations. Recent years have seen an increased interest in MonteCarlo-based methods for smoothing, often involving particle filters. One suchmethod is to approximate filter distributions with a particle filter, and thento simulate backwards on the trellis of particles using a backward kernel. Weshow that by supplementing this procedure with a Metropolis-Hastings stepdeciding whether to accept a proposed trajectory or not, one obtains a Markovchain Monte Carlo scheme whose stationary distribution is the exact smoothingdistribution. We also show that in this procedure, backward sampling can bereplaced by backward smoothing, which effectively means averaging over allpossible trajectories. In an example we compare these approaches to a similarone recently proposed by Andrieu, Doucet and Holenstein, and show that the newmethods can be more efficient in terms of precision (inverse variance) percomputation time.

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