# Automatic forward model parameterization with Bayesian inference of conformational populations

## TL;DR

The paper introduces new methods to improve computational models by optimizing parameters using Bayesian inference, enhancing agreement between simulations and experimental data.

## Contribution

The study introduces two novel methods for optimizing forward model parameters using Bayesian inference and variational minimization.

## Key findings

- Treating forward model parameters as nuisance parameters improves Bayesian inference accuracy.
- Variational minimization of the BICePs score optimizes parameters for better model validation.
- The approach successfully refines Karplus relation parameters for J-coupling constants in human ubiquitin.

## Abstract

To quantify how well theoretical predictions of structural ensembles agree with experimental measurements, we depend on the accuracy of forward models (FMs). These models are computational frameworks that generate observable quantities from molecular configurations based on empirical relationships linking specific molecular properties to experimental measurements. Bayesian Inference of Conformational Populations (BICePs) is a reweighting algorithm that reconciles simulated ensembles with ensemble-averaged experimental observations, even when such observations are sparse and/or noisy. This is achieved by sampling the posterior distribution of conformational populations under experimental restraints as well as sampling the posterior distribution of uncertainties due to random and systematic error. In this study, we enhance the algorithm for the refinement of empirical FM parameters. We introduce and evaluate two novel methods for optimizing FM parameters. The first method treats FM parameters as nuisance parameters, integrating over them in the full posterior distribution. The second method employs variational minimization of a quantity called the BICePs score that reports the free energy of “turning on” the experimental restraints. This technique, coupled with improved likelihood functions for handling experimental outliers, facilitates force field validation and optimization, as illustrated in recent studies [R. M. Raddi et al., J. Chem. Theory Comput. 21, 5880–5889 (2025) and R. M. Raddi and V. A. Voelz, “Automated optimization of force field parameters against ensemble-averaged measurements with Bayesian inference of conformational populations,” arXiv:2402.11169 (2024)]. Using this approach, we refine parameters that modulate the Karplus relation, crucial for accurate predictions of J-coupling constants based on dihedral angles (ϕ) between interacting nuclei. We validate this approach first with a toy model system and then for human ubiquitin, predicting six sets of Karplus parameters for JHNHα3, JHαC′3, JHNCβ3, JHNC′3, JC′Cβ3, and JC′C′3. Finally, we demonstrate that our framework naturally generalizes optimization to any differentiable FM, such as those constructed with neural networks. This approach provides a promising direction for training and validating neural network-based FMs.