Geometric ergodicity of Metropolis algorithms
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.
scientific article; zbMATH DE number 1792507