A splitting method to reduce MCMC variance

We explore whether splitting and killing methods can improve the accuracy of Markov chain Monte Carlo (MCMC) estimates of rare event probabilities, and we make three contributions. First, we prove that "weighted ensemble" is the only splitting and killing method that provides asymptotically consistent estimates when combined with MCMC. Second, we prove a lower bound on the asymptotic variance of weighted ensemble's estimates. Third, we give a constructive proof and numerical examples to show that weighted ensemble can approach this optimal variance bound, in many cases reducing the variance of MCMC estimates by multiple orders of magnitude.

[1]  Eric F Darve,et al.  Analysis of the accelerated weighted ensemble methodology , 2013 .

[2]  P. Moral,et al.  Genealogical particle analysis of rare events , 2005, math/0602525.

[3]  Gerardo Rubino,et al.  Rare Event Simulation using Monte Carlo Methods , 2009 .

[4]  Robert J. Webber,et al.  Unifying Sequential Monte Carlo with Resampling Matrices , 2019, 1903.12583.

[5]  van Aernout Enter.  Statistical Mechanics, A Short Treatise , 2000 .

[6]  Daniel M. Zuckerman,et al.  Extensive Evaluation of Weighted Ensemble Strategies for Calculating Rate Constants and Binding Affinities of Molecular Association/Dissociation Processes , 2019, bioRxiv.

[7]  Ute Dreher,et al.  Statistical Mechanics Rigorous Results , 2016 .

[8]  Galin L. Jones.  On the Markov chain central limit theorem , 2004, math/0409112.

[9]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[10]  David Aristoff,et al.  Analysis and optimization of weighted ensemble sampling , 2016, ESAIM: Mathematical Modelling and Numerical Analysis.

[11]  G. Huber,et al.  Weighted-ensemble Brownian dynamics simulations for protein association reactions. , 1996, Biophysical journal.

[12]  Daniel M. Zuckerman,et al.  Optimizing Weighted Ensemble Sampling of Steady States , 2018, Multiscale Model. Simul..

[13]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[14]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[15]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[16]  A. Sokal.  Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms , 1997 .

[17]  Daniel M Zuckerman,et al.  Weighted Ensemble Simulation: Review of Methodology, Applications, and Software. , 2017, Annual review of biophysics.

[18]  Aaron R. Dinner,et al.  Trajectory Stratification of Stochastic Dynamics , 2016, SIAM Rev..

[19]  F. Cérou,et al.  Adaptive Multilevel Splitting for Rare Event Analysis , 2007 .

[20]  Paul Glasserman,et al.  Multilevel Splitting for Estimating Rare Event Probabilities , 1999, Oper. Res..

[21]  P. Grassberger.  Pruned-enriched Rosenbluth method: Simulations of θ polymers of chain length up to 1 000 000 , 1997 .

[22]  Reuven Y. Rubinstein,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.

[23]  Daniel M. Zuckerman,et al.  Accelerated estimation of long-timescale kinetics from weighted ensemble simulation via non-Markovian "microbin" analysis. , 2020, Journal of chemical theory and computation.

[24]  Jeremy Copperman,et al.  Accelerated Estimation of Long-timescale Kinetics by Combining Weighted Ensemble Simulation with Markov Model “Microstates” using Non-Markovian Theory , 2019, Biophysical Journal.

[25]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[26]  David Aristoff,et al.  An ergodic theorem for weighted ensemble , 2019, ArXiv.

[27]  Ernest K. Ryu,et al.  Computing reaction rates in bio-molecular systems using discrete macro-states , 2012, 1307.0763.

[28]  Sarwar Hussain,et al.  Studying rare events using forward-flux sampling: Recent breakthroughs and future outlook. , 2019, The Journal of chemical physics.

[29]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[30]  Daniel M Zuckerman,et al.  The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures. , 2008, The Journal of chemical physics.

[31]  S. Fahy.  Quantum Monte Carlo Methods , 1996 .

[32]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[33]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[34]  Lillian T. Chong,et al.  A Minimal, Adaptive Binning Scheme for Weighted Ensemble Simulations , 2020, bioRxiv.

[35]  Bin W. Zhang,et al.  Steady-state simulations using weighted ensemble path sampling. , 2009, The Journal of chemical physics.

[36]  Arnaud Guyader,et al.  Adaptive multilevel splitting: Historical perspective and recent results. , 2019, Chaos.

[37]  A. W. Rosenbluth,et al.  MONTE CARLO CALCULATION OF THE AVERAGE EXTENSION OF MOLECULAR CHAINS , 1955 .

[38]  Eric Moulines,et al.  Comparison of resampling schemes for particle filtering , 2005, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005..

[39]  Anthony C. Davison,et al.  Bootstrap Methods and Their Application , 1998 .