The Research On Convergence Rates Of Time-inhomogeneous Markov Chains And MCMC Algorithms

The MCMC algorithms was born in the early 1950 s,and Metropolis introduced the Markov chain into the Monte Carlo method,which was later generalized by Hastings to make it more general.Among them,the most widely used is the M-H algorithm and Gibbs sampling.The MCMC algorithms mainly relies on the convergence of the Markov chain simulation,that is,whether the Markov chain constructed by it converges,and how fast the convergence speed is.Therefore,qualitative analysis and quantitative analysis for the convergence rates of Markov chains are particularly important.In recent decades,many scholars have studied the quantitative bound of convergence rates of Markov chains under relevant conditions and used them in many algorithms and models.Based on the convergence estimation of homogeneous markov chains,this paper generalizes relevant theories to the case of non-homogeneous markov chains.The main work is as follows:First,a brief introduction to the basic theory of Markov chains,including stationary distribution and invariant measure,etc.Then this paper introduces the reversibility,geometric ergodicity,and total variation distance and coupling method of Markov chains that will be used in subsequent chapters,and summarizes and compare the conclusions of exponential convergence and subgeometric convergence in the estimation of convergence rates of homogeneous Markov chains.Secondly,it expounds the Monte Carlo method based on Markov chains and the M-H algorithm and Gibbs algorithm widely used in MCMC method.Finally,under the weaken minorization condition and the post-deformation drift condition,starting from the Wasserstein distance,the convergence rates of the timeinhomogeneous Markov chain is quantitatively analyzed under the coupling method.Then,under the analytic method,the quantitative bound of the convergence rates of time-inhomogeneous markov chains is obtained by defining some metrics and norms.The results of the two methods are also compared and analyzed,and they are used in the M-H algorithm and the nonhomogeneous model.