On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method (Communications in Computational Physics, 2014)
In Markov Chain Monte Carlo (MCMC) simulations, thermal equilibria quantitiesare estimated by ensemble average over a sample set containing a large numberof correlated samples. These samples are selected in accordance with the probabilitydistribution function, known from the partition function of equilibrium state. As thestochastic error of the simulation results is significant, it is desirable to understand thevariance of the estimation by ensemble average, which depends on the sample size(i.e., the total number of samples in the set) and the sampling interval (i.e., cycle numberbetween two consecutive samples). Although large sample sizes reduce the variance,they increase the computational cost of the simulation. For a given CPU time, thesample size can be reduced greatly by increasing the sampling interval, while havingthe corresponding increase in variance be negligible if the original sampling intervalis very small. In this work, we report a few general rules that relate the variance withthe sample size and the sampling interval. These results are observed and confirmednumerically. These variance rules are derived for the MCMC method but are also validfor the correlated samples obtained using other Monte Carlo methods. The main contributionof this work includes the theoretical proof of these numerical observationsand the set of assumptions that lead to them.