  Bayesian models have become very popular over the last years in several
fields such as signal processing, statistics, and machine learning. Bayesian
inference requires the approximation of complicated integrals involving
posterior distributions. For this purpose, Monte Carlo (MC) methods, such as
Markov Chain Monte Carlo and importance sampling algorithms, are often
employed. In this work, we introduce the theory and practice of a Compressed MC
(C-MC) scheme to compress the statistical information contained in a set of
random samples. In its basic version, C-MC is strictly related to the
stratification technique, a well-known method used for variance reduction
purposes. Deterministic C-MC schemes are also presented, which provide very
good performance. The compression problem is strictly related to the moment
matching approach applied in different filtering techniques, usually called as
Gaussian quadrature rules or sigma-point methods. C-MC can be employed in a
distributed Bayesian inference framework when cheap and fast communications
with a central processor are required. Furthermore, C-MC is useful within
particle filtering and adaptive IS algorithms, as shown by three novel schemes
introduced in this work. Six numerical results confirm the benefits of the
introduced schemes, outperforming the corresponding benchmark methods. A
related code is also provided.
