  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.
