We consider Markov chain Monte Carlo algorithms which combine Gibbs updates with Metropolis-Hastings updates, resulting in a conditional Metropolis-Hastings sampler (CMH sampler). We develop conditions under which the CMH sampler will be geometrically or uniformly ergodic. We illustrate our results by analysing a CMH sampler used for drawing Bayesian inferences about the entire sample path of a diffusion process, based only upon discrete observations.
"Convergence of conditional Metropolis-Hastings samplers." Adv. in Appl. Probab. 46 (2) 422 - 445, June 2014. https://doi.org/10.1239/aap/1401369701