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.

💡 Research Summary

The paper addresses the smoothing problem in state‑space models, which consists of computing the conditional distribution of the latent state trajectory given a sequence of observations, or expectations of functionals with respect to that distribution. For non‑linear, non‑Gaussian, or discrete‑state models the exact smoothing distribution is intractable because it involves integrals over a space whose dimensionality grows with the number of observations. Particle filters provide a Monte‑Carlo approximation of the filtering distributions, and a common strategy is to run a backward kernel on the particle trellis (forward‑filtering backward‑sampling, FFBS) to generate whole trajectories. However, FFBS only yields an approximation to the smoothing distribution and suffers from sampling variance.

The authors propose two complementary enhancements to the standard FFBS scheme. First, they embed a Metropolis–Hastings (MH) acceptance step into the backward‑sampling procedure, creating a “Metropolised” backward sampler. The particle filter’s approximation to the filtering distribution is used as the proposal for the whole trajectory, and the MH acceptance probability is computed from the product of forward and backward transition densities. This turns the trajectory generation into a Markov‑chain Monte‑Carlo (MCMC) kernel whose invariant distribution is exactly the true smoothing distribution. Consequently, the Metropolisation corrects the bias introduced by the particle approximation while preserving the computational structure of FFBS.

Second, the authors replace the stochastic backward‑sampling step with a deterministic backward‑smoothing operation. Instead of selecting a single ancestor at each time step, they compute a weighted average over all possible ancestors, where the weights are proportional to the forward particle weights multiplied by the backward transition probabilities. This is a Rao‑Blackwellisation of the trajectory estimator: by analytically integrating out the discrete ancestor indices, the variance of the estimator is reduced without additional sampling. The resulting algorithm, called the Rao‑Blackwellised Metropolised Particle Smoother (RB‑MPS), combines the exactness of the MH correction with the variance reduction of Rao‑Blackwellisation.

The paper provides rigorous theoretical results. The authors define the transition kernel of the Metropolised particle smoother, prove that it satisfies detailed balance with respect to the smoothing distribution, and establish geometric ergodicity under standard regularity conditions. For the Rao‑Blackwellised version they show that the estimator is the conditional expectation of the trajectory given the particle filter output, guaranteeing that its mean‑square error is never larger than that of the plain Metropolised sampler.

Computational complexity is analysed: the Metropolised step adds only the evaluation of an acceptance ratio (which is O(T) for a trajectory of length T) to the usual O(NT) cost of the particle filter, where N is the number of particles. The Rao‑Blackwellised version requires an additional O(N²) pass to compute the backward smoothing weights, but this cost is amortised across all trajectories because the same weights are reused for every MH iteration.

Empirical evaluation is carried out on two benchmark models. The first is a non‑linear stochastic difference equation with a sinusoidal transition and Gaussian observation noise. The second is a switching linear dynamical system where a discrete mode governs the linear dynamics. For each model the authors compare three algorithms: (i) the standard FFBS, (ii) the Andrieu‑Doucet‑Holenstein (ADH) particle smoother, and (iii) the proposed Metropolised and Rao‑Blackwellised Metropolised smoothers. Results are reported in terms of mean‑square error (MSE) of the estimated state means and of the computational time required to achieve a given precision. The Metropolised smoother consistently outperforms ADH in terms of precision per unit time, and the Rao‑Blackwellised variant yields the lowest MSE, especially in long time series (hundreds of observations) and higher‑dimensional state spaces (dimension ≥10). The variance reduction achieved by Rao‑Blackwellisation is evident from the narrower confidence intervals and the faster convergence of empirical averages.

In summary, the paper makes two key contributions: (1) a Metropolis‑adjusted backward‑sampling scheme that turns particle‑filter‑based trajectory generation into an exact MCMC sampler for the smoothing distribution, and (2) a Rao‑Blackwellised version that replaces stochastic ancestor selection with deterministic averaging, thereby reducing estimator variance. Both methods retain the scalability of particle filters while providing theoretical guarantees of convergence to the true smoothing distribution. The authors suggest future work on adaptive particle numbers, parallel implementation, and extensions to high‑dimensional models where the backward‑smoothing step could be approximated using low‑rank or variational techniques.