Mean-field particle methods

Mean-field particle methods are a broad class of interacting type Monte Carlo algorithms for simulating from a sequence of probability distributions satisfying a nonlinear evolution equation.[1][2][3][4] These flows of probability measures can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depends on the distributions of the current random states.[1][2] A natural way to simulate these sophisticated nonlinear Markov processes is to sample a large number of copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methods these mean-field particle techniques rely on sequential interacting samples. The terminology mean-field reflects the fact that each of the samples (a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with the empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. In other words, starting with a chaotic configuration based on independent copies of initial state of the nonlinear Markov chain model, the chaos propagates at any time horizon as the size the system tends to infinity; that is, finite blocks of particles reduces to independent copies of the nonlinear Markov process. This result is called the propagation of chaos property.[5][6][7] The terminology "propagation of chaos" originated with the work of Mark Kac in 1976 on a colliding mean-field kinetic gas model.[8]

  1. ^ a b Kolokoltsov, Vassili (2010). Nonlinear Markov processes. Cambridge Univ. Press. p. 375.
  2. ^ a b Del Moral, Pierre (2013). Mean field simulation for Monte Carlo integration. Monographs on Statistics & Applied Probability. Vol. 126. ISBN 9781466504059.
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  8. ^ Kac, Mark (1976). Probability and Related Topics in Physical Sciences. Topics in Physical Sciences. American Mathematical Society, Providence, Rhode Island.