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Asymptotic variance for random walk metropolis chains in high dimensions : logarithmic growth via the Poisson equation
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Mijatović, Aleksandar and Vogrinc, Jure (2019) Asymptotic variance for random walk metropolis chains in high dimensions : logarithmic growth via the Poisson equation. Advances in Applied Probability, 51 (4). pp. 994-1026. doi:10.1017/apr.2019.40
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WRAP-asymptotic-variance-random-walk-metropolis-chains-high-dimensions-Vogrinc-2019.pdf - Accepted Version - Requires a PDF viewer. Download (892Kb) | Preview |
Official URL: https://doi.org/10.1017/apr.2019.40
Abstract
There are two ways of speeding up Markov chain Monte Carlo algorithms: (a) construct more complex samplers that use gradient and higher-order information about the target and (b) design a control variate to reduce the asymptotic variance. While the efficiency of (a) as a function of dimension has been studied extensively, this paper provides the first results linking the efficiency of (b) with dimension. Specifically, we construct a control variate for a d-dimensional random walk Metropolis chain with an independent, identically distributed target using the solution of the Poisson equation for the scaling limit in [30]. We prove that the asymptotic variance of the corresponding estimator is bounded above by a multiple of over the spectral gap of the chain. The proof hinges on large deviations theory, optimal Young’s inequality and Berry–Esseen-type bounds. Extensions of the result to non-product targets are discussed.
| Item Type: | Journal Article | |||||||||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||||||||
| Divisions: | Faculty of Science > Statistics | |||||||||||||||
| Library of Congress Subject Headings (LCSH): | Random walks (Mathematics), Asymptotic expansions | |||||||||||||||
| Journal or Publication Title: | Advances in Applied Probability | |||||||||||||||
| Publisher: | Applied Probability Trust | |||||||||||||||
| ISSN: | 0001-8678 | |||||||||||||||
| Official Date: | December 2019 | |||||||||||||||
| Dates: |
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| Volume: | 51 | |||||||||||||||
| Number: | 4 | |||||||||||||||
| Page Range: | pp. 994-1026 | |||||||||||||||
| DOI: | 10.1017/apr.2019.40 | |||||||||||||||
| Status: | Peer Reviewed | |||||||||||||||
| Publication Status: | Published | |||||||||||||||
| Publisher Statement: | Accepted for publication by the Applied Probability Trust (http://www.appliedprobability.org) in Advances in Applied Probability, 51 (4). | |||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||||||||
| Copyright Holders: | © Applied Probability Trust 2019 | |||||||||||||||
| RIOXX Funder/Project Grant: |
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