Constructing sampling schemes via coupling: Markov semigroups and optimal transport

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Title: Constructing sampling schemes via coupling: Markov semigroups and optimal transport
Authors: Nüsken, N
Pavliotis, GA
Item Type: Journal Article
Abstract: In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.
Issue Date: 26-Mar-2019
Date of Acceptance: 18-Jan-2019
URI: http://hdl.handle.net/10044/1/69635
DOI: https://dx.doi.org/10.1137/18m119896x
ISSN: 2166-2525
Publisher: Society for Industrial and Applied Mathematics
Start Page: 324
End Page: 382
Journal / Book Title: SIAM/ASA Journal on Uncertainty Quantification
Volume: 7
Issue: 1
Copyright Statement: © 2019 Society for Industrial and Applied Mathematics and American Statistical Association.
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Funder's Grant Number: EP/J009636/1
EP/L020564/1
EP/P031587/1
Keywords: Science & Technology
Physical Sciences
Mathematics, Interdisciplinary Applications
Physics, Mathematical
Mathematics
Physics
sampling
optimal transport
particle methods
Markov semigroups
MCMC
CHAIN MONTE-CARLO
INVARIANT MEASURE
CONVERGENCE
EQUATIONS
THEOREM
Publication Status: Published
Online Publication Date: 2019-03-26
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics



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