Summarizing posterior distributions in signal decomposition problems when the number of components is unknown

TL;DR

This paper introduces a new method to summarize complex Bayesian posterior distributions in signal decomposition problems with unknown components, using a parametric approximation and a Stochastic EM algorithm driven by RJ-MCMC sampling.

Contribution

It proposes a novel approach to approximate variable-dimensional posteriors with a parametric distribution, enabling effective summarization in signal decomposition tasks.

Findings

The method successfully summarizes posterior distributions in sinusoid detection.

It effectively estimates component-specific parameters from complex posteriors.

The approach is demonstrated on a fundamental signal processing example.

Abstract

This paper addresses the problem of summarizing the posterior distributions that typically arise, in a Bayesian framework, when dealing with signal decomposition problems with unknown number of components. Such posterior distributions are defined over union of subspaces of differing dimensionality and can be sampled from using modern Monte Carlo techniques, for instance the increasingly popular RJ-MCMC method. No generic approach is available, however, to summarize the resulting variable-dimensional samples and extract from them component-specific parameters. We propose a novel approach to this problem, which consists in approximating the complex posterior of interest by a "simple"---but still variable-dimensional---parametric distribution. The distance between the two distributions is measured using the Kullback-Leibler divergence, and a Stochastic EM-type algorithm, driven by the RJ-MCMC sampler, is proposed to estimate the parameters. The proposed algorithm is illustrated on the fundamental signal processing example of joint detection and estimation of sinusoids in white Gaussian noise.