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The Metropolis Hastings algorithm: introduction and optimal scaling of the transient phase

By Benjamin Jourdain

Appears in collections : 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, Ecoles de recherche

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).

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Citation data

  • DOI 10.24350/CIRM.V.19199403
  • Cite this video Jourdain, Benjamin (19/07/2017). The Metropolis Hastings algorithm: introduction and optimal scaling of the transient phase. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19199403
  • URL https://dx.doi.org/10.24350/CIRM.V.19199403

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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