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Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity

Kuntz, J. and Ottobre, M. and Stuart, A. M. (2014) Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity. . (Submitted)

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The Random Walk Metropolis (RWM) algorithm is a Metropolis- Hastings MCMC algorithm designed to sample from a given target distribution \pi with Lebesgue density on R^N. 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 \pi. 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 [RGG97, RR98]; in particular in [RGG97] 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 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. The target measures that we consider 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), as it is in stationarity. 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 [KOS16].

Item Type:Report or Paper (Discussion Paper)
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Additional Information: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.
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
Classification Code:MSC 2010 subject classifications: Primary 60J22; secondary 60J20, 60H10
Record Number:CaltechAUTHORS:20161221-115035181
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73084
Deposited By: Linda Taddeo
Deposited On:21 Dec 2016 20:33
Last Modified:21 Dec 2016 20:33

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