On convergence rates of Gibbs samplers for uniform distributions
Let \(R\in{\mathbf R}^d\) be a bounded open connected region in \(d\)-dimensional Euclidean space, and \({\mathbf X}^{(0)}\) be some random variable taking values in \(R\). The random-scan Gibbs sampler proceeds as follows. Given a point \({\mathbf X}^{(n)}\in{\mathbf R}^d\), it chooses \(I_{n+1} \in\{1,2,\dots,d\}\) uniformly at random. It then chooses \({\mathbf X}^{(n+1)}\) uniformly from the one-dimensional set \(\{(X_1^{(n)}, \dots,X_{I-1}^{(n)}, y,X_{I+1}^{(n)}, \dots, X_d^{(n)})\); \(y\in{\mathbf R}^1\}\cap R\). This process is repeated for \(n=0,1, \dots\). The random-scan Gibbs sampler algorithm implicitly defines Markov chain transition probabilities \(L({\mathbf X}^{(n+1)}\mid {\mathbf X}^{(n)})\). A Markov chain with state space \({\mathfrak R}\) and stationary distribution \(\pi(\cdot)\) is called geometrically ergodic if there is \(\rho<1\), a subset \({\mathfrak R}_0 \subseteq {\mathfrak R}\) with \(\pi({\mathfrak R}_0)=1\) and \(M:{\mathfrak R}_0\to{\mathbf R}^1\) such that the total variation distance \(\|L({\mathbf X}^{(n)} \mid{\mathbf X}^{(0)} =x_0)-\pi(\cdot) \|\leq M(x_0)\rho^n\), \(n\in{\mathbf N}\), \(x_0\in{\mathfrak R}_0\). The chain is uniformly ergodic if it is geometrically ergodic with \(M\) constant. The authors consider the random-scan Gibbs sampler applied to the uniform distribution on \(R\). It is shown that the convergence properties of the Gibbs sampler depend greatly on the smoothness of the boundary of \(R\). For sufficiently smooth boundaries the sampler is uniformly ergodic. For slightly less regular regions \(R\) the sampler is still geometrically ergodic. However, for jagged boundaries the sampler could fail to even be geometrically ergodic.
scientific article; zbMATH DE number 1319913