Geometric Ergodicity & Scanning Strategies For Two-Component Gibbs Samplers

In any Markov chain Monte Carlo analysis, rapid convergence of the chain to its target probability distribution is of practical and theoretical importance. A chain that converges at a geometric rate is geometrically ergodic. In this paper, we explore geometric ergodicity for two-component Gibbs samplers which, under a chosen scanning strategy, evolve by combining one-at-a-time updates of the two components. We compare convergence behaviors between and within three such strategies: composition, random sequence scan, and random scan. Our main results are twofold. First, we establish that if the Gibbs sampler is geometrically ergodic under any one of these strategies, so too are the others. Further, we establish a simple and verifiable set of sufficient conditions for the geometric ergodicity of the Gibbs samplers. Our results are illustrated using two examples.

💡 Research Summary

This paper investigates the convergence speed of two‑component Gibbs samplers under three common scanning schemes: composition (deterministic order), random‑sequence scan (RSS) and random scan (RS). The authors first formalize each scheme as a Markov transition kernel—(P_{C}), (P_{RSS}) and (P_{RS})—and recall that geometric ergodicity means the total‑variation distance to the target distribution decays at a geometric rate, i.e., (|P^{n}(x,\cdot)-\pi|_{TV}\le M(x)\rho^{n}) for some (\rho<1).

The central theoretical contribution is a strategy‑equivalence theorem: if a two‑component Gibbs sampler is geometrically ergodic under any one of the three kernels, it is geometrically ergodic under the other two. The proof rests on two observations. First, the RSS kernel can be expressed as the average of the two possible composition kernels, while the RS kernel is a convex combination of the one‑step updates of each component. Second, the spectral radius of each kernel on the space of centered functions is bounded by the same constant less than one whenever one kernel satisfies a drift‑minorization condition. Consequently, the geometric rate (\rho) transfers across the schemes.

To make the result usable, the authors provide verifiable sufficient conditions for geometric ergodicity that are independent of the scanning order. They require a drift function (V\ge1) and constants (\lambda\in(0,1)), (b<\infty) such that
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