Generalized Rejection Sampling Schemes and Applications in Signal Processing

TL;DR

This paper introduces generalized rejection sampling schemes that adaptively approximate complex target distributions, significantly improving sampling efficiency in signal processing applications involving non-standard and multimodal probability densities.

Contribution

The paper presents new methods for obtaining upper bounds and adaptively computing proposal densities, culminating in the generalized adaptive rejection sampling (GARS) algorithm for efficient sampling.

Findings

GARS converges to the target distribution efficiently.

The methods handle multimodal and non-log-concave distributions.

Sampling efficiency is significantly improved in practical signal processing problems.

Abstract

Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques, such as Markov chain Monte Carlo (MCMC) and particle filters, have become very popular in signal processing over the last years. However, in many problems of practical interest these techniques demand procedures for sampling from probability distributions with non-standard forms, hence we are often brought back to the consideration of fundamental simulation algorithms, such as rejection sampling (RS). Unfortunately, the use of RS techniques demands the calculation of tight upper bounds for the ratio of the target probability density function (pdf) over the proposal density from which candidate samples are drawn. Except for the class of log-concave target pdf's, for which an efficient algorithm exists, there are no general methods to analytically determine this bound, which has to be derived from scratch for each specific case. In this paper, we introduce new schemes for (a) obtaining upper bounds for likelihood functions and (b) adaptively computing proposal densities that approximate the target pdf closely. The former class of methods provides the tools to easily sample from a posteriori probability distributions (that appear very often in signal processing problems) by drawing candidates from the prior distribution. However, they are even more useful when they are exploited to derive the generalized adaptive RS (GARS) algorithm introduced in the second part of the paper. The proposed GARS method yields a sequence of proposal densities that converge towards the target pdf and enable a very efficient sampling of a broad class of probability distributions, possibly with multiple modes and non-standard forms.