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Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity

Kuntz, Juan and Ottobre, Michela and Stuart, Andrew M. (2019) Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity. Annales De l'Institut Henri Poincaré - Probabilitiés et Statistiques, 55 (3). pp. 1599-1648. ISSN 0246-0203. https://resolver.caltech.edu/CaltechAUTHORS:20161221-115035181

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Abstract

The Random Walk Metropolis (RWM) algorithm is a Metropolis–Hastings Markov Chain Monte Carlo algorithm designed to sample from a given target distribution π^N with Lebesgue density on R^N. Like any other Metropolis–Hastings algorithm, RWM constructs a Markov chain by randomly proposing a new position (the “proposal move”), which is then accepted or rejected according to a rule which makes the chain reversible with respect to π^N. When the dimension N is large, a key question is to determine the optimal scaling with N of the proposal variance: if the proposal variance is too large, the algorithm will reject the proposed moves too often; if it is too small, the algorithm will explore the state space too slowly. Determining the optimal scaling of the proposal variance gives a measure of the cost of the algorithm as well. One approach to tackle this issue, which we adopt here, is to derive diffusion limits for the algorithm. Such an approach has been proposed in the seminal papers (Ann. Appl. Probab. 7 (1) (1997) 110–120; J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 (1) (1998) 255–268). In particular, in (Ann. Appl. Probab. 7 (1) (1997) 110–120) the authors derive a diffusion limit for the RWM algorithm under the two following assumptions: (i) the algorithm is started in stationarity; (ii) the target measure π^N is in product form. The present paper considers the situation of practical interest in which both assumptions (i) and (ii) are removed. That is (a) we study the case (which occurs in practice) in which the algorithm is started out of stationarity and (b) we consider target measures which are in non-product form. Roughly speaking, we consider target measures that admit a density with respect to Gaussian; such measures arise in Bayesian nonparametric statistics and in the study of conditioned diffusions. We prove that, out of stationarity, the optimal scaling for the proposal variance is O(N^(−1)), as it is in stationarity. In this optimal scaling, a diffusion limit is obtained and the cost of reaching and exploring the invariant measure scales as O(N). Notice that the optimal scaling in and out of stationatity need not be the same in general, and indeed they differ e.g. in the case of the MALA algorithm (Stoch. Partial Differ. Equ. Anal Comput. 6 (3) (2018) 446–499). More importantly, our diffusion limit is given by a stochastic PDE, coupled to a scalar ordinary differential equation; such an ODE gives a measure of how far from stationarity the process is and can therefore be taken as an indicator of convergence. In this sense, this paper contributes understanding to the old-standing problem of monitoring convergence of MCMC algorithms.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1214/18-AIHP929DOIArticle
https://arxiv.org/abs/1405.4896arXivDiscussion Paper
ORCID:
AuthorORCID
Ottobre, Michela0000-0002-8725-4278
Additional Information:© 2019 Association des Publications de l’Institut Henri Poincaré. Received: 3 September 2016; Revised: 19 December 2017; Accepted: 28 August 2018; First available in Project Euclid: 25 September 2019. Supported by ERC and EPSRC. We are extremely grateful to the anonimous referee for his/her careful reading, for spotting mistakes in an earlier version and for comments that helped improving the paper.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)UNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Subject Keywords:Markov Chain Monte Carlo, Random Walk Metropolis algorithm, diffusion limit, optimal scaling
Issue or Number:3
Classification Code:MSC 2010 subject classifications: Primary 60J22; secondary 60J20, 60H10
Record Number:CaltechAUTHORS:20161221-115035181
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20161221-115035181
Official Citation:Kuntz, Juan; Ottobre, Michela; Stuart, Andrew M. Diffusion limit for the random walk Metropolis algorithm out of stationarity. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1599-1648. doi:10.1214/18-AIHP929. https://projecteuclid.org/euclid.aihp/1569398880
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73084
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:21 Dec 2016 20:33
Last Modified:10 Oct 2019 21:51

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