Improving individual committor estimates and data efficiency in reaction coordinate tests with the empirical Bayes method

Abstract

Several rare event methods for computing rates, for assessing collective variables, and for testing mechanistic hypotheses involve estimating the committor at an ensemble of configurations. Each committor is a probability that the dynamics arrive at the product state before reaching the reactant state, i.e., a Bernoulli parameter. A distribution of committors reflects both configuration-to-configuration differences in committors and binomial sampling error in the estimation of individual committors. We use the empirical Bayes method to simultaneously improve the committor ensemble description and the accuracy of the individual committor estimates. We illustrate the method for an exactly analogous and tangible situation using a sample of randomly deformed thumbtacks. Each thumbtack has a different probability of landing “point up” when tossed, but the individual Bernoulli parameters and the ensemble of Bernoulli parameters are both unknown. We toss 40 tacks 20 times each to obtain the usual maximum likelihood estimate for each Bernoulli parameter. We then use the empirical Bayes method to construct a prior from the ensemble of individual estimates and obtain new estimates by maximizing the posterior for each tack. The empirical Bayes method definitively improves the individual Bernoulli parameter estimates. For a radical chain walking reaction that occurs in pyrolysis of polyethylene, we use the committor analysis with the empirical Bayes approach to show that an antisymmetric stretch (“Jorgensen”) coordinate is more accurate than a simple bond distance (“coordination number”) coordinate.