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Asymptotic Analysis of the Random-walk Metropolis Algorithm on Ridged Densities

Beskos, A; Roberts, G; Thiery, A; Pillai, N; (2018) Asymptotic Analysis of the Random-walk Metropolis Algorithm on Ridged Densities. Annals of Applied Probability , 28 (5) pp. 2966-3001. 10.1214/18-AAP1380. Green open access

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Abstract

We study the asymptotic behavior of the Random-Walk Metropolis algorithm on ‘ridged’ probability densities where most of the probability mass is distributed along some key directions. Such class of probability measures arise in various applied contexts including for instance Bayesian inverse problems where the posterior measure concentrates on a manifold when the noise variance goes to zero. When the target measure concentrates on a linear manifold, we derive analytically a diffusion limit for the Random-Walk Metropolis Markov chain as the scale parameter goes to zero. In contrast to the existing works on scaling limits, our limiting Stochastic Differential Equation does not in general have a constant diffusion coefficient. Our results show that in some cases, the usual practice of adapting the step-size to control the acceptance probability might be sub-optimal as the optimal acceptance probability is zero (in the limit).

Type: Article
Title: Asymptotic Analysis of the Random-walk Metropolis Algorithm on Ridged Densities
Open access status: An open access version is available from UCL Discovery
DOI: 10.1214/18-AAP1380
Publisher version: https://doi.org/10.1214/18-AAP1380
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Manifold, random-walk metropolis, generator, diffusion limit
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/1474802
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