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

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