CIRM - Videos & books Library - The Metropolis Hastings algorithm: introduction and optimal scaling of the transient phase

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Authors : Jourdain, Benjamin (Author of the conference)
CIRM (Publisher )

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Abstract : We first introduce the Metropolis-Hastings algorithm. We then consider the Random Walk Metropolis algorithm on $R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known that, in the limit $n$ tends to infinity, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain. We generalize this result when the initial distribution is not the target probability measure. The obtained diffusive limit is the solution to a stochastic differential equation nonlinear in the sense of McKean. We prove convergence to equilibrium for this equation. We discuss practical counterparts in order to optimize the variance of the proposal distribution to accelerate convergence to equilibrium. Our analysis confirms the interest of the constant acceptance rate strategy (with acceptance rate between 1/4 and 1/3).

MSC Codes :
60F17 - Functional limit theorems; invariance principles
60G09 - Exchangeability
60G50 - Sums of independent random variables; random walks
60J10 - Markov chains (discrete-time Markov processes on discrete state spaces)
60J22 - Computational methods in Markov chains
60J60 - Diffusion processes
65C05 - Monte Carlo methods
65C40 - Computational Markov chains (numerical analysis)

Additional resources :
http://smai.emath.fr/cemracs/cemracs17/Slides/jourdain.pdf

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2017-07-19_Jourdain.mp4

Information on the Event

Event Title : CEMRACS - Summer school: Numerical methods for stochastic models: control, uncertainty quantification, mean-field / CEMRACS - École d'été : Méthodes numériques pour équations stochastiques : contrôle, incertitude, champ moyen
Event Organizers : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme
Dates : 17/07/17 - 25/08/17
Event Year : 2017
Event URL : http://conferences.cirm-math.fr/1556.html

Citation Data

DOI : 10.24350/CIRM.V.19199403
Cite this video as: Jourdain, Benjamin (2017). The Metropolis Hastings algorithm: introduction and optimal scaling of the transient phase. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19199403
URI : http://dx.doi.org/10.24350/CIRM.V.19199403

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Bibliography

Jourdain, B., Lelièvre, T., & Miasojedow, B. (2015). Optimal scaling for the transient phase of the random walk Metropolis algorithm: the mean-field limit. The Annals of Applied Probability, 25(4), 2263-2300 - http://dx.doi.org/10.1214/14-AAP1048

Jourdain, B., Lelièvre, T., & Miasojedow, B. (2014). Optimal scaling for the transient phase of Metropolis Hastings algorithms: the longtime behavior. Bernoulli, 20(4), 1930-1978 - http://dx.doi.org/10.3150/13-BEJ546

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