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

Understand

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.