# Group Importance Sampling for Particle Filtering and MCMC

## TL;DR

This paper introduces Group Importance Sampling (GIS), a novel approach that consolidates weighted samples into a single representative sample, enhancing Monte Carlo methods like particle filtering and MCMC with new algorithms and theoretical insights.

## Contribution

The work presents the theory of GIS, interprets existing algorithms under this framework, and develops two new MCMC techniques based on GIS for improved Bayesian inference.

## Key findings

- GIS provides a unified way to summarize weighted samples.
- New MCMC algorithms based on GIS outperform benchmarks in experiments.
- The proposed methods are effective in diverse applications like Gaussian Processes and satellite data analysis.

## Abstract

Bayesian methods and their implementations by means of sophisticated Monte Carlo techniques have become very popular in signal processing over the last years. Importance Sampling (IS) is a well-known Monte Carlo technique that approximates integrals involving a posterior distribution by means of weighted samples. In this work, we study the assignation of a single weighted sample which compresses the information contained in a population of weighted samples. Part of the theory that we present as Group Importance Sampling (GIS) has been employed implicitly in different works in the literature. The provided analysis yields several theoretical and practical consequences. For instance, we discuss the application of GIS into the Sequential Importance Resampling framework and show that Independent Multiple Try Metropolis schemes can be interpreted as a standard Metropolis-Hastings algorithm, following the GIS approach. We also introduce two novel Markov Chain Monte Carlo (MCMC) techniques based on GIS. The first one, named Group Metropolis Sampling method, produces a Markov chain of sets of weighted samples. All these sets are then employed for obtaining a unique global estimator. The second one is the Distributed Particle Metropolis-Hastings technique, where different parallel particle filters are jointly used to drive an MCMC algorithm. Different resampled trajectories are compared and then tested with a proper acceptance probability. The novel schemes are tested in different numerical experiments such as learning the hyperparameters of Gaussian Processes, two localization problems in a wireless sensor network (with synthetic and real data) and the tracking of vegetation parameters given satellite observations, where they are compared with several benchmark Monte Carlo techniques. Three illustrative Matlab demos are also provided.