Collective sampling through a Metropolis-Hastings like method: kinetic theory and numerical experiments
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The classical Metropolis-Hastings algorithm provides a simple method to construct a Markov Chain with an arbitrary stationary measure. In order to implement Monte Carlo methods, an elementary approach would be to duplicate this algorithm as many times as desired. Following the ideas of Population Monte Carlo methods, we propose to take advantage of the number of duplicates to increase the efficiency of the naive approach. Within this framework, each chain is seen as the evolution of a single particle which interacts with the others. In this article, we propose a simple and efficient interaction mechanism and an analytical framework which ensures that the particles are asymptotically independent and identically distributed according to an arbitrary target law. This approach is also supported by numerical simulations showing better convergence properties compared to the classical Metropolis-Hastings algorithm.
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