Paper detail

Particle Filtering and Smoothing Using Windowed Rejection Sampling

&#34;Particle methods&#34; are sequential Monte Carlo algorithms, typically involving importance sampling, that are used to estimate and sample from joint and marginal densities from a collection of a, presumably increasing, number of random variables. In particular, a particle filter aims to estimate the current state $X_{n}$ of a stochastic system that is not directly observable by estimating a posterior distribution $π(x_{n}|y_{1},y_{2}, \ldots, y_{n})$ where the $\{Y_{n}\}$ are observations related to the $\{X_{n}\}$ through some measurement model $π(y_{n}|x_{n})$. A particle smoother aims to estimate a marginal distribution $π(x_{i}|y_{1},y_{2}, \ldots, y_{n})$ for $1 \leq i < n$. Particle methods are used extensively for hidden Markov models where $\{X_{n}\}$ is a Markov chain as well as for more general state space models. Existing particle filtering algorithms are extremely fast and easy to implement. Although they suffer from issues of degeneracy and &#34;sample impoverishment&#34;, steps can be taken to minimize these problems and overall they are excellent tools for inference. However, if one wishes to sample from a posterior distribution of interest, a particle filter is only able to produce dependent draws. Particle smoothing algorithms are complicated and far less robust, often requiring cumbersome post-processing, &#34;forward-backward&#34; recursions, and multiple passes through subroutines. In this paper we introduce an alternative algorithm for both filtering and smoothing that is based on rejection sampling &#34;in windows&#34; . We compare both speed and accuracy of the traditional particle filter and this &#34;windowed rejection sampler&#34; (WRS) for several examples and show that good estimates for smoothing distributions are obtained at no extra cost.