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Bayesian Multistate Bennett Acceptance Ratio Methods

Xinqiang Ding

TL;DR

BayesMBAR extends MBAR by placing a prior on the free energies $F$ and deriving the posterior $p(F|Y,X)$ from the reverse logistic regression likelihood, enabling posterior-based estimates and uncertainty quantification for free-energy calculations. A uniform prior recovers MBAR at the MAP, while Gaussian-process priors encode smoothness over collective coordinates and are learned via Bayesian evidence, with posterior moments computed by NUTS when needed. Across two- and three-state harmonic oscillator benchmarks and a phenol hydration example, BayesMBAR improves uncertainty estimates in small-sample regimes and, with informative priors, yields more accurate free energy estimates than MBAR; the approach offers principled prior integration and uncertainty quantification at the cost of additional computation. The authors provide an open-source implementation and discuss extensions to incorporate diverse prior knowledge from related calculations or experiments, highlighting practical impact for robust free-energy calculations in chemistry and physics.

Abstract

The multistate Bennett acceptance ratio (MBAR) method is a prevalent approach for computing free energies of thermodynamic states. In this work, we introduce BayesMBAR, a Bayesian generalization of the MBAR method. By integrating configurations sampled from thermodynamic states with a prior distribution, BayesMBAR computes a posterior distribution of free energies. Using the posterior distribution, we derive free energy estimations and compute their associated uncertainties. Notably, when a uniform prior distribution is used, BayesMBAR recovers the MBAR's result but provides more accurate uncertainty estimates. Additionally, when prior knowledge about free energies is available, BayesMBAR can incorporate this information into the estimation procedure by using non-uniform prior distributions. As an example, we show that, by incorporating the prior knowledge about the smoothness of free energy surfaces, BayesMBAR provides more accurate estimates than the MBAR method. Given MBAR's widespread use in free energy calculations, we anticipate BayesMBAR to be an essential tool in various applications of free energy calculations.

Bayesian Multistate Bennett Acceptance Ratio Methods

TL;DR

BayesMBAR extends MBAR by placing a prior on the free energies and deriving the posterior from the reverse logistic regression likelihood, enabling posterior-based estimates and uncertainty quantification for free-energy calculations. A uniform prior recovers MBAR at the MAP, while Gaussian-process priors encode smoothness over collective coordinates and are learned via Bayesian evidence, with posterior moments computed by NUTS when needed. Across two- and three-state harmonic oscillator benchmarks and a phenol hydration example, BayesMBAR improves uncertainty estimates in small-sample regimes and, with informative priors, yields more accurate free energy estimates than MBAR; the approach offers principled prior integration and uncertainty quantification at the cost of additional computation. The authors provide an open-source implementation and discuss extensions to incorporate diverse prior knowledge from related calculations or experiments, highlighting practical impact for robust free-energy calculations in chemistry and physics.

Abstract

The multistate Bennett acceptance ratio (MBAR) method is a prevalent approach for computing free energies of thermodynamic states. In this work, we introduce BayesMBAR, a Bayesian generalization of the MBAR method. By integrating configurations sampled from thermodynamic states with a prior distribution, BayesMBAR computes a posterior distribution of free energies. Using the posterior distribution, we derive free energy estimations and compute their associated uncertainties. Notably, when a uniform prior distribution is used, BayesMBAR recovers the MBAR's result but provides more accurate uncertainty estimates. Additionally, when prior knowledge about free energies is available, BayesMBAR can incorporate this information into the estimation procedure by using non-uniform prior distributions. As an example, we show that, by incorporating the prior knowledge about the smoothness of free energy surfaces, BayesMBAR provides more accurate estimates than the MBAR method. Given MBAR's widespread use in free energy calculations, we anticipate BayesMBAR to be an essential tool in various applications of free energy calculations.
Paper Structure (4 sections, 18 equations, 3 figures, 5 tables)

This paper contains 4 sections, 18 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Results
  4. Conclusion and Discussion

Figures (3)

  • Figure 1: Probability densities of the posterior distribution (solid line) of $\Delta F$ and the approximate Gaussian distribution (dashed line) used by the asymptotic analysis for the two harmonic oscillator system with $n_1 = n_2 = 18$.
  • Figure 2: Probability density and samples of the posterior distribution of $F_2 - F_1$ and $F_3 - F_1$ for the three harmonic oscillators with $n = 18$. (a) Contours are the logarithm of the posterior distribution density. Dots are a subset of samples drawn from the posterior distribution using the NUTS sampler. (b and c) The first 300 samples of $F_2 - F_1$ and $F_3 - F_1$ drawn from the posterior distribution using the NUTS sampler.
  • Figure 3: Free energy estimates of all states for computing the hydration free energy of phenol. SD_ F (BayesMBAR), SD_ F (asymptotic), and SD_ F (bootstrap) are the average of the uncertainty estimates using BayesMBAR, the asymptotic analysis, and the bootstrap method, respectively, when $n = 5$. SD_ F (true) is the true uncertainty when $n = 5$. $F_\mathrm{ref}$ is the MBAR estimate computed using all configurations sampled from all repeats.