Geometric ergodicity of Metropolis algorithms
type: Item
instance of: scholarly article
title: Geometric ergodicity of Metropolis algorithms
zbMATH Open document ID: 0997.60070
DOI: 10.1016/S0304-4149(99)00082-4
author: Søren Fiig Jarner
author: Ernst Hansen
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: Let \(\Pi \) be a probability measure on \(\mathbb R^{k}\) with a strictly positive continuous density \(\pi \). The random-walk-based Metropolis algorithm is a method to construct a Markov chain \(\{X_{n}\}\) on \(\mathbb R^{k}\) with the invariant measure \(\Pi \). The chain \(\{X_{n}\}\) has the transition kernel \(P\) given by \(P(x, dy) = p(x,y) dy + r(x) d\delta _{x}(y)\), where \(\delta _{x}\) is the point mass at \(x\), \(p(x,y) = \alpha (x,y) q(|x-y|)\) for \(x\neq y\), \(p(x,x)=0\), \(\alpha (x,y) = \min \{1,\pi ^{-1} (x)\pi (y)\}\), \(r(x) = \int _{\mathbb R^{k}} (1-\alpha (x,y))q(|x-y|) dy\) and the function \(q\geq 0\) satisfies \(\int _{\mathbb R^{k}} q(|x|) dx =1\), \(q\geq c\) in some neighbourhood of 0 for a constant \(c>0\). Necessary and/or sufficient conditions for the chain \(\{X_{n}\}\) to be geometrically ergodic are studied. Let us quote two typical results: If \(\int _{\mathbb R^{k}} |x|q(|x|) dx <\infty \) and \(P\) is geometrically ergodic, then \(\int _{\mathbb R^{k}} \exp (s|x|)\pi (x) dx <\infty \) for some \(s>0\). If \(\pi \in C^1(\mathbb R^{k})\) is super-exponential, i.e. \(\langle |x|^{-1}x,\nabla \log \pi (x)\rangle \to -\infty \) as \(|x|\to \infty \) and if \(\limsup _{|x|\to \infty } \langle |x|^{-1}x, |\nabla \pi (x)|^{-1}\nabla \pi (x)\rangle <0\), then \(P\) is geometrically ergodic.
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 60J22
zbMATH DE Number: 1792507
zbMATH Keywords: Markov chain Monte Carlo
zbMATH Keywords: Metropolis algorithm
zbMATH Keywords: geometric ergodicity
zbMATH Keywords: super-exponential densities
MaRDI profile type: Publication
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full work available at URL: s0304-4149(99)00082-4
OpenAlex ID: W2017874618
description: scientific article; zbMATH DE number 1792507