Combining multiple interface set path ensembles with MBAR reweighting

TL;DR

The paper addresses the challenge of integrating transition-path ensembles generated from multiple interface sets defined by different CVs. It introduces a $\text{MultiSet-}\text{MBAR}$ framework that reweights and merges trajectories across $M$ interface sets into a single unbiased Reweighted Path Ensemble, with trajectory weights determined by the maxima reached in each set. The approach is validated on a 2D double-well model and demonstrated on a Host–Guest AIMMD-TIS system, showing improved crossing-probability estimates and more accurate free-energy representations compared with single-set MBAR or independent rescalings. The method enables iterative interface optimization and data reuse, offering practical benefits for mechanistic insight and efficiency in complex rare-event simulations.

Abstract

We introduce a method to compute the reweighted path ensemble by combining transition interface sampling simulations conditioned on different collective variables. The approach is based on the Multistate Bennett Acceptance Ratio (MBAR) methodology applied to entire trajectories. Illustrating the technique with simple 2D potential models and a more complex host-guest system, we show that the statistics can significantly improve compared to a straightforward combination.

Figures (6)

• Figure 1: Illustration of the intersection of $\lambda$ and $\mu$ interfaces and the joint initial interface defined as the first interface that trajectories cross.
• Figure 2: (a) The double well potential with illustration of the two sets of interfaces along $\lambda$ and $\mu$ for the forward ensembles. (b) The weights assigned to each trajectory based on its maximum $k^{max}_{\lambda}$ and $k_{\mu}^{max}$ in the space of $\lambda$ and $\mu$ interfaces. (c) The crossing probability for the forward and backward ensembles computed through two-set MBAR and through a single set benchmark calculation, showing that the two-set procedure recovers the same crossing probability. (d) The free energy of the RPE overlayed with the potential surface. (e) The free energy error relative to the potential energy surface.
• Figure 3: (top row) Estimated crossing probability $P_A(\lambda_B|\lambda_1)$ using $\lambda(x) = x$ as a function of the number of sets $M$ being combined, for different number of paths per interface, $N$, for both the reactive matching method and the MultiSet-MBAR approach. Results are compared to a benchmark computed from $N=10000$ paths sampled along the $x$ interface set. The MultiSet-MBAR estimate converges smoothly to the benchmark as more sets are added, while the independent reactive matching estimate initially diverges for small $N$ and only aligns at larger sample sizes. (middle row) Logarithmic deviation of the estimated crossing probabilities from the benchmark value as a function of the number of sets $M$ being combined, for different number of paths per interface, $N$, for both the independent reactive matching method and the MultiSet-MBAR approach. The MultiSet-MBAR method consistently reduces deviation as more data is included, demonstrating improved convergence. In contrast, the reactive matching approach shows increased deviation for low $N$, especially as additional sets are added. (bottom row) Relative statistical error of the crossing probability estimates, defined as $\sigma_{\text{bs}}/\mathbb{E}[\log P_A(\lambda_B|\lambda_1)]_{\text{bs}}$, as a function of the number of sets $M$ being combined, for different number of paths per interface, $N$, for both the independent reactive matching method and the MultiSet-MBAR approach. The MultiSet-MBAR method exhibits a consistent reduction in relative error, approximately following a $1/\sqrt{M}$ trend with increasing number of sets. In contrast, the reactive matching approach shows a plateau in error for small $N$, only improving when the sample size becomes sufficiently large.
• Figure 4: Weighted mean absolute error (MAE) between RPE free energy and potential energy surface, weighted by the density of trajectory configurations per bin, as the number of sets increases for N=10, 50,100,1000, using bootstrapping with 100 repeated resampling of $N$ trajectories to get a statistical error.
• Figure 5: Forward crossing probability $P_A(q(x|\theta_{\mathrm{RPE}_1}) \mid q_A)$ evaluated using reweighted path ensembles from successive TIS iterations on a committor model $q(x|\theta)$ for a host-guest system. Independent RPEs constructed from TIS simulations using $q(x|\theta_{\mathrm{TPS}})$ (green) and $q(x|\theta_{\mathrm{RPE}_1})$ (grey) are compared with their joint reweighting using MultiSet-MBAR (blue). Also shown are independent RPE rescaling strategies based on reactive matching (red) and flux matching (brown). Due to the extremely large statistical uncertainty associated with flux matching, its error bars are omitted for clarity, the relative error is reported in the text.