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On the geometric ergodicity of hybrid samplers. (English) Zbl 1028.65002

Markov chain Monte Carlo algorithm can also be seen as schemes for drawing samples from an ergodic Markov chain with a given stationary distribution \(p\) on a state space \(X\). When the dimension of the state space is large, \(X\) can be written as a product of lower-dimensional state spaces, \(X= X_1\times X_2\times\cdots\times X_d\), and a Markov transition kernel \(P\) can be constructed on \(X\) having the stationary distribution \(p\) by combining the kernel operators \(P_i\) that act on \(X_i\). The deterministic scan Gibbs sampler is an example of such a strategy. This method is referred in several papers as ‘variable-at-a-time Metropolis-Hastings’ or ‘Metropolis-within-Gibbs’ algorithms.
The present paper considers \(C= (P_1,P_2,\dots, P_d)\) a collection of Markov kernels on a state space \(X= \mathbb{R}^d\), the random scan hybrid sampler for \(C\) is defined as the arithmetical mean of the kernels \(P_i\), \(i= 1,\dots, d\), where each operator \(P_i\) arises from a symmetric random-walk Metropolis algorithm on the \(i\)th coordinate, and the resulted procedure is referred as ‘random-scan symmetric random walk Metropolis’ (RSM) algorithm. The RSM algorithm performs a Metropolis step at a time, as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which performs a transition on all coordinates at once. One of the assumptions considered in the papers studying recently the RSM algorithm is expressed in terms of the maximal curvature of all the geodesic curves on the contour manifold, a condition that is difficult to be checked even for \(d= 2\).
The novelty of this paper is that the authors succeed to show that geometric ergodicity holds with essentially no conditions on the geometry of the contour manifold. Section 2 of the paper presents a sufficient RSM condition for the geometric ergodicity of the RSM algorithm on \(\mathbb{R}^d\) for subexponential densities, in terms of \(V\)-uniform ergodicity of the kernel \(P\). The analysis of six analytical examples support the proposed approach with \(V\)-uniformly ergodic functions. Section 3 extends the investigation of geometric ergodicity to densities that are log-concave in the tails, in particular when the target density is exponential. A necessary condition for \(V\)-uniform ergodicity of the RSM algorithm is examined, and other three analytical examples illustrate the proposed theoretical framework.

MSC:

65C05 Monte Carlo methods
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65C40 Numerical analysis or methods applied to Markov chains
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