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Bayesian methods and their implementations by means of sophisticated MonteCarlo techniques have become very popular in signal processing over the lastyears. Importance Sampling (IS) is a well-known Monte Carlo technique thatapproximates integrals involving a posterior distribution by means of weightedsamples. In this work, we study the assignation of a single weighted samplewhich compresses the information contained in a population of weighted samples.Part of the theory that we present as Group Importance Sampling (GIS) has beenemployed implicitly in different works in the literature. The provided analysisyields several theoretical and practical consequences. For instance, we discussthe application of GIS into the Sequential Importance Resampling framework andshow that Independent Multiple Try Metropolis schemes can be interpreted as astandard Metropolis-Hastings algorithm, following the GIS approach. We alsointroduce two novel Markov Chain Monte Carlo (MCMC) techniques based on GIS.The first one, named Group Metropolis Sampling method, produces a Markov chainof sets of weighted samples. All these sets are then employed for obtaining aunique global estimator. The second one is the Distributed ParticleMetropolis-Hastings technique, where different parallel particle filters arejointly used to drive an MCMC algorithm. Different resampled trajectories arecompared and then tested with a proper acceptance probability. The novelschemes are tested in different numerical experiments such as learning thehyperparameters of Gaussian Processes, two localization problems in a wirelesssensor network (with synthetic and real data) and the tracking of vegetationparameters given satellite observations, where they are compared with severalbenchmark Monte Carlo techniques. Three illustrative Matlab demos are alsoprovided.

Group Importance Sampling for Particle Filtering and MCMC