Relabeling and Summarizing Posterior Distributions in Signal Decomposition Problems when the Number of Components is Unknown

TL;DR

This paper introduces VAPoRS, a novel method for relabeling and summarizing variable-dimensional posterior distributions in Bayesian signal decomposition, effectively handling unknown component numbers using a parametric approximation and stochastic EM algorithm.

Contribution

The paper presents VAPoRS, a new approach that approximates complex variable-dimensional posteriors with a parametric model, enabling effective relabeling and summarization.

Findings

VAPoRS successfully relabels components in sinusoid detection.

VAPoRS effectively summarizes astrophysical particle counting data.

The method improves interpretation of variable-dimensional Bayesian posteriors.

Abstract

This paper addresses the problems of relabeling and summarizing posterior distributions that typically arise, in a Bayesian framework, when dealing with signal decomposition problems with an 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, named Variable-dimensional Approximate Posterior for Relabeling and Summarizing (VAPoRS), to this problem, which consists in approximating the posterior distribution 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. Two signal decomposition problems are considered, to show the capability of VAPoRS both for relabeling and for summarizing variable dimensional posterior distributions: the classical problem of detecting and estimating sinusoids in white Gaussian noise on the one hand, and a particle counting problem motivated by the Pierre Auger project in astrophysics on the other hand.