Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition

It is common practice in Markov chain Monte Carlo to update the simulation one variable (or sub-block of variables) at a time, rather than conduct a single full-dimensional update. When it is possible to draw from each full-conditional distribution associated with the target this is just a Gibbs sampler. Often at least one of the Gibbs updates is replaced with a Metropolis-Hastings step, yielding a Metropolis-Hastings-within-Gibbs algorithm. Strategies for combining component-wise updates include composition, random sequence and random scans. While these strategies can ease MCMC implementation and produce superior empirical performance compared to full-dimensional updates, the theoretical convergence properties of the associated Markov chains have received limited attention. We present conditions under which some component-wise Markov chains converge to the stationary distribution at a geometric rate. We pay particular attention to the connections between the convergence rates of the various component-wise strategies. This is important since it ensures the existence of tools that an MCMC practitioner can use to be as confident in the simulation results as if they were based on independent and identically distributed samples. We illustrate our results in two examples including a hierarchical linear mixed model and one involving maximum likelihood estimation for mixed models.

💡 Research Summary

The paper addresses a fundamental yet under‑explored aspect of modern Markov chain Monte Carlo (MCMC): the convergence behavior of component‑wise updating schemes. In many high‑dimensional Bayesian models it is impractical to propose a full‑dimensional move at each iteration; instead practitioners update one variable or a block of variables at a time. When each full‑conditional distribution can be sampled directly this reduces to a Gibbs sampler, but more often at least one conditional is intractable and must be handled by a Metropolis–Hastings (MH) step, yielding a Metropolis‑within‑Gibbs algorithm.

The authors categorize three common ways of combining component updates: (i) Composition (deterministic ordering of the component kernels), (ii) Random Sequence (a new random permutation of the components at each iteration), and (iii) Random Scan (select a single component at random according to a fixed probability vector). While these strategies are known to improve empirical mixing, rigorous results on their convergence rates have been scarce.

The core contribution is a set of sufficient conditions guaranteeing uniform ergodicity (total‑variation distance bounded uniformly over the state space) and geometric ergodicity (total‑variation distance decays at a geometric rate) for the overall Markov chain generated by any of the three strategies. The analysis rests on the classical drift‑and‑minorization framework. The authors show that if each component kernel (K_i) admits a common drift function (V\ge 1), a small set (C), and constants (\lambda\in(0,1)), (\varepsilon\in(0,1)) such that

\