Exact Convergence Rate Analysis of the Independence Metropolis-Hastings Algorithms

A well-known difficult problem regarding Metropolis-Hastings algorithms is to get sharp bounds on their convergence rates. Moreover, different initializations may have different convergence rates so a uniform upper bound may be too conservative to be used in practice. In this paper, we study the convergence properties of the Independence Metropolis-Hastings (IMH) algorithms on both general and discrete state spaces. Under mild conditions, we derive the exact convergence rate and prove that different initializations of the IMH algorithm have the same convergence rate. In particular, we get the exact convergence speed for IMH algorithms on general state spaces under certain conditions. Connections with the Random Walk Metropolis-Hastings (RWMH) algorithm are also discussed, which solve the conjecture proposed by Atchade and Perron using a counterexample.