Exact Approximation of Rao-Blackwellised Particle Filters

Abstract

Particle methods are a category of Monte Carlo algorithms that have become popular for performing inference in non-linear non-Gaussian state-space models. The class of "RaoBlackwellised" particle filters exploits the analytic marginalisation that is possible for some statespace models to reduce the variance of the Monte Carlo estimates. Despite being applicable to only a restricted class of state-space models, such as conditionally linear Gaussian models, these algorithms have found numerous applications. In scenarios where no such analytical integration is possible, it has recently been proposed in Chen et al. [2011] to use "local" particle filters to carry out this integration numerically. We propose here an alternative approach also relying on "local" particle filters which is more broadly applicable and has attractive theoretical properties. Proof-of-concept simulation results are presented.

Bayesian Conditional Monte Carlo Algorithms for Nonlinear Time-Series State Estimation

“…Now, in a sequential filtering context, is typically only known up to a constant; however a sequential application of this technique ensures that the (marginal) target distribution deduced from the approximated weights is still . The idea has already been applied to spatial RB [6], [7] and, as we shall see (see in particular Section III-D), can be adapted to the CMC problem discussed in this paper.…”

“…In other models, it may be possible to approximate by using numerical approximations of and of ; however, due to the spatial structure of the decomposition of , approximating in (8) involves propagating numerical approximations over time. Finally, recent contributions [6], [7] propose to approximate the integral in (8) via a local MC method which also leads to an approximation of the importance weights in (8).…”

“…In Bayesian filtering the idea is either known as marginalized or Rao-Blackwellized Particle Filters (RB-PF) and has been applied so far to a spatial partition of the state vectors [3]- [7]; by spatial we mean that each vector is split as (where and are respectively the "linear" and "non-linear" components of vector ), so that matrix is partitioned into two block rows. In this paper we propose another class of RB-PF; the main difference is that our partitioning of is now temporal rather than spatial, i.e., that matrix is split into two (block) columns, and .…”

A Bayesian Uncertainty Analysis for Nonignorable Nonresponse

“…This is in essence an approximated RBPF, where we use local Monte Carlo approximations instead of analytical solutions. This idea was proposed in [27,46] and also in [41] in the context of noise adaptive PF. A related but slightly different formulation is considered in [1,11].…”

“…Consequently one cannot implement an RBPF for this case. For such cases, in recent times, different approximations of the RBPF schemes have been envisaged in the literature (see e.g., [1,11,27,41,46]). The basic idea again is to decompose the whole state space into two (interacting) parts 6 and split the filtering problem into two nested sub-problems.…”