Analysis of two-component Gibbs samplers using the theory of two projections

The theory of two projections is utilized to study two-component Gibbs samplers. Through this theory, previously intractable problems regarding the asymptotic variances of two-component Gibbs samplers are reduced to elementary matrix algebra exercises. It is found that in terms of asymptotic variance, the two-component random-scan Gibbs sampler is never much worse, and could be considerably better than its deterministic-scan counterpart, provided that the selection probability is appropriately chosen. This is especially the case when there is a large discrepancy in computation cost between the two components. The result contrasts with the known fact that the deterministic-scan version has a faster convergence rate. A modified version of the deterministic-scan sampler that accounts for computation cost behaves similarly to the random-scan version. As a side product, some general formulas for characterizing the convergence rate of a possibly non-reversible or time-inhomogeneous Markov chain in an operator theoretic framework are developed.