## Statistical Science

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

#### Abstract

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.

#### Article information

Source
Statist. Sci., Volume 28, Number 3 (2013), 360-375.

Dates
First available in Project Euclid: 28 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.ss/1377696941

Digital Object Identifier
doi:10.1214/13-STS423

Mathematical Reviews number (MathSciNet)
MR3135537

Zentralblatt MATH identifier
1331.60151

#### Citation

Johnson, Alicia A.; Jones, Galin L.; Neath, Ronald C. Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity under Mixing and Composition. Statist. Sci. 28 (2013), no. 3, 360--375. doi:10.1214/13-STS423. https://projecteuclid.org/euclid.ss/1377696941

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#### Supplemental materials

• Supplementary material: Supplementary material for “Component-Wise Markov Chain Monte Carlo: Uniform and Geometric Ergodicity Under Mixing and Composition”. This supplemental article includes all technical details and proofs for the above results.