Bayesian models have become very popular over the last years in several fields such as signal processing, statistics and machine learning. Bayesian inference needs the approximation of complicated integrals involving the posterior distribution. For this purpose, Monte Carlo (MC) methods, such as Markov Chain Monte Carlo (MCMC) and Importance Sampling (IS) algorithms, are often employed. In this work, we introduce theory and practice of a Compressed MC (C-MC) scheme, in order to compress the information contained in a could of samples. CMC is particularly useful in a distributed Bayesian inference framework, when cheap and fast communications with a central processor are required. 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 moment matching approach applied in different filtering methods, often known as Gaussian quadrature rules or sigma-point methods. The connections to herding algorithms and quasi-Monte Carlo perspective are also discussed. Numerical results confirm the benefit of the introduced schemes, outperforming the corresponding benchmark methods.
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