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The Metropolis Hastings algorithm: introduction and optimal scaling of the transient phase
Auteurs : Jourdain, Benjamin (Auteur de la Conférence)
CIRM (Editeur )
CIRM (Editeur )
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- 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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Résumé :
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).
Codes MSC : 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)
Ressources complémentaires :
http://smai.emath.fr/cemracs/cemracs17/Slides/jourdain.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 30/07/17 Date de captation : 19/07/17 Collection : Research School Format : MP4 Durée : 00:58:51 Domaine : Probability & Statistics ; Numerical Analysis & Scientific Computing Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2017-07-19_Jourdain.mp4 
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Informations sur la rencontre
Nom de la rencontre : 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 moyenOrganisateurs de la rencontre : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme Dates : 17/07/17 - 25/08/17 Année de la rencontre : 2017 URL Congrès : http://conferences.cirm-math.fr/1556.html Citation Data
DOI : 10.24350/CIRM.V.19199403Cite 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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