Multistate Bennett acceptance ratio can be used to reweigh one or more samples from a guided diffusion trajectory

Summary

Multistate Bennett acceptance ratio can be used to reweight one or more weighted samples from a guided diffusion trajectory (1). It does not work unless the samples are pre-weighted; (1) use the steering importance weights from Feynman-Kac guidance.

Details

Originally presented by (2).

Given $K$ states, each with a potential energy function $u_k$, and $N_k$ samples drawn from state $k$, MBAR finds the free energies $f_k$ that satisfy the self-consistent equations:

$${\hat{f}}_k=-\ln \sum _{n=1}^N\frac{\exp (-u_k(x_n))}{{\sum }_{l=1}^K{N}_l\exp (f_l-u_l(x_n))}$$

These are solved iteratively. The solution is the maximum likelihood estimator for the free energies, which is a key theoretical strength — it’s provably optimal given the data you have.

Once you have the $f_k$, you can compute the importance weight $W_n$ for any sample $x_n$ under a target distribution $t$:

$W_n\propto \frac{\exp (-u_t(x_n))}{{\sum }_l{N}_l\exp (f_l-u_l(x_n))}$

1.

Xie Y, Winkler L, Sun L, Lewis S, Foster AE, Luna JJ, et al. Enhanced Diffusion Sampling: Efficient Rare Event Sampling and Free Energy Calculation with Diffusion Models. 2026; Available from: https://arxiv.org/abs/2602.16634

2.

Shirts MR, Chodera JD. Statistically optimal analysis of samples from multiple equilibrium states. The Journal of Chemical Physics. 2008;129(12). Available from: https://doi.org/10.1063/1.2978177