  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.
