Markov chain Monte Carlo (MCMC) algorithms are generally regarded as the gold standard technique for Bayesian inference. They theoretically well-understood and conceptually simple to apply in practice. The drawback of MCMC is that performing exact inference requires all data be processed at each iteration algorithm. For large datasets, computational cost can prohibitive, which has led recent de...

In this paper we shall briefly review a few Markov Chain Monte Carlo methods for simulating closed systems described by canonical ensembles. We cover both Boltzmann and non-Boltzmann sampling techniques. The Metropolis algorithm is a typical example of Boltzmann Monte Carlo method. We discuss the time-symmetry of the Markov chain generated by Metropolis like algorithms that obey detailed balanc...

We develop a martingale approximation approach to studying the limiting behavior of quadratic forms of Markov chains. We use the technique to examine the asymptotic behavior of lag-window estimators in time series and we apply the results to Markov Chain Monte Carlo simulation. As another illustration, we use the method to derive a central limit theorem for U-statistics with varying kernels.