Exact Convergence Rate Analysis of the Independent Metropolis-Hastings Algorithms

A well-known difficult problem regarding Metropolis-Hastings algorithms is to get sharp bounds on their convergence rates. Moreover, a fundamental but often overlooked problem in Markov chain theory is to study the convergence rates for different initializations. In this paper, we study the two issues mentioned above of the Independent Metropolis-Hastings (IMH) algorithms on both general and discrete state spaces. We derive the exact convergence rate and prove that the IMH algorithm's different deterministic initializations have the same convergence rate. Surprisingly, under mild conditions, we get the exact convergence speed for IMH algorithms on general state spaces, which is the first `exact convergence' result for general state space MCMC algorithms to the author's best knowledge. Connections with the Random Walk Metropolis-Hastings (RWMH) algorithm are also discussed, which solve a conjecture proposed by Atchad\'{e} and Perron \cite{atchade2007geometric} using a counterexample.