Exact convergence 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. We get the exact convergence speed for IMH algorithms on general state spaces.