Editing Markov chain Monte Carlo (section)

=== Interacting particle methods ===
Interacting MCMC methodologies are a class of [[mean-field particle methods]] for obtaining [[Pseudo-random number sampling|random samples]] from a sequence of probability distributions with an increasing level of sampling complexity.<ref name="dp13">{{cite book|last = Del Moral|first = Pierre|title = Mean field simulation for Monte Carlo integration|year = 2013|publisher = Chapman & Hall/CRC Press |url = http://www.crcpress.com/product/isbn/9781466504059|pages = 626}}</ref> These probabilistic models include path space state models with increasing time horizon, posterior distributions w.r.t. sequence of partial observations, increasing constraint level sets for conditional distributions, decreasing temperature schedules associated with some Boltzmann–Gibbs distributions, and many others. In principle, any Markov chain Monte Carlo sampler can be turned into an interacting Markov chain Monte Carlo sampler. These interacting Markov chain Monte Carlo samplers can be interpreted as a way to run in parallel a sequence of Markov chain Monte Carlo samplers. For instance, interacting [[simulated annealing]] algorithms are based on independent Metropolis–Hastings moves interacting sequentially with a selection-resampling type mechanism. In contrast to traditional Markov chain Monte Carlo methods, the precision parameter of this class of interacting Markov chain Monte Carlo samplers is ''only'' related to the number of interacting Markov chain Monte Carlo samplers. These advanced particle methodologies belong to the class of Feynman–Kac particle models,<ref name="dp04">{{cite book|last = Del Moral|first = Pierre|title = Feynman–Kac formulae. Genealogical and interacting particle approximations|year = 2004|publisher = Springer |url = https://www.springer.com/mathematics/probability/book/978-0-387-20268-6|pages = 575}}</ref><ref name="dmm002">{{cite book|last1 = Del Moral|first1 = Pierre|last2 = Miclo|first2 = Laurent|contribution = Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering|title=Séminaire de Probabilités XXXIV |editor=Jacques Azéma |editor2=Michel Ledoux |editor3=Michel Émery |editor4=Marc Yor|series = Lecture Notes in Mathematics|date = 2000|volume = 1729|pages = 1–145|url = http://archive.numdam.org/ARCHIVE/SPS/SPS_2000__34_/SPS_2000__34__1_0/SPS_2000__34__1_0.pdf|doi = 10.1007/bfb0103798|isbn = 978-3-540-67314-9}}</ref> also called Sequential Monte Carlo or [[particle filter]] methods in [[Bayesian inference]] and [[signal processing]] communities.<ref name=":3">{{Cite journal|title = Sequential Monte Carlo samplers | doi=10.1111/j.1467-9868.2006.00553.x|volume=68|issue = 3|year=2006|journal=Journal of the Royal Statistical Society. Series B (Statistical Methodology)|pages=411–436 | last1 = Del Moral | first1 = Pierre|arxiv=cond-mat/0212648| s2cid=12074789}}</ref> Interacting Markov chain Monte Carlo methods can also be interpreted as a mutation-selection [[Genetic algorithm|genetic particle algorithm]] with Markov chain Monte Carlo mutations.