Minorization Conditions and Convergence Rates for Markov Chain Monte Carlo

General methods are provided for analyzing the convergence of discrete-time, general state-space Markov chains, such as those used in stochastic simulation algorithms including the Gibbs sampler. The methods provide rigorous, a priori bounds on how long these simulations should be run to give satisfactory results. Results are applied to two models of the Gibbs sampler: a bivariate normal model, and a hierarchical Poisson model (with gamma conditionals). The methods use the notion of minorization conditions for Markov chains.