  To quantify how well theoretical predictions of structural ensembles agree
with experimental measurements, we depend on the accuracy of forward models.
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 forward model (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 (Raddi et al.
2023, 2024). Using this approach, we refine parameters that modulate the
Karplus relation, crucial for accurate predictions of J-coupling constants
based on dihedral angles between interacting nuclei. We validate this approach
first with a toy model system, and then for human ubiquitin, predicting six
sets of Karplus parameters. This approach, which does not rely on predetermined
parameters, enhances predictive accuracy and can be used for many applications.
