TY - JOUR

T1 - On stochastic error and computational efficiency of the Markov Chain Monte Carlo method

AU - Li, Jun

AU - Vignal, Philippe

AU - Sun, Shuyu

AU - Calo, Victor M.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: This work was supported in part by the King Abdullah University of Science and Technology (KAUST) Center for Numerical Porous Media. In addition, S. Sun would also like to acknowledge the support of this study by a research award from King Abdulaziz City for Science and Technology (KACST) through a project entitled "Study of Sulfur Solubility using Thermodynamics Model and Quantum Chemistry".

PY - 2015/6/3

Y1 - 2015/6/3

N2 - In Markov Chain Monte Carlo (MCMC) simulations, thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance 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 number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for theMCMCmethod but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them. © 2014 Global-Science Press.

AB - In Markov Chain Monte Carlo (MCMC) simulations, thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance 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 number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for theMCMCmethod but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them. © 2014 Global-Science Press.

UR - http://hdl.handle.net/10754/563327

UR - https://www.cambridge.org/core/product/identifier/S1815240600005636/type/journal_article

UR - http://www.scopus.com/inward/record.url?scp=84903159666&partnerID=8YFLogxK

U2 - 10.4208/cicp.110613.280214a

DO - 10.4208/cicp.110613.280214a

M3 - Article

VL - 16

SP - 467

EP - 490

JO - Communications in Computational Physics

JF - Communications in Computational Physics

SN - 1815-2406

IS - 2

ER -