Bayesian Sampling of Structural Ensembles: The Role of Ensemble-Counting Measures
Abstract
Structural ensemble refinement is widely used to integrate molecular simulations with experimental measurements. While most applications focus on the maximum-a-posteriori (MAP) ensemble, Bayesian sampling of the posterior distribution can provide uncertainty estimates and posterior averages for arbitrary observables. A notable step in this direction was introduced by the Bayesian Energy Landscape Tilting (BELT) framework, where sampling is performed on a family of maximum-entropy ensembles parametrized by Lagrange multipliers. Here, we show that Bayesian sampling in this setting requires an explicit choice of ensemble-counting measure. In particular, the flat measure in Lagrange-multiplier space used in the original BELT formulation leads to a posterior distribution that is formally non-normalizable for finite reference trajectories. We propose the Jeffreys measure as an invariant ensemble-counting prescription, restoring normalizability in the finite-sample situations considered here, and providing a consistent definition of posterior averages. Using both an analytically tractable Gaussian model and maximum-entropy refinement of RNA oligomer simulations, we compare different ensemble-counting measures and show that they can significantly affect Bayesian estimates. The resulting methodology has been implemented in the \texttt{MDRefine} software package.
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BibTeX
@misc{gilardoni2026bayesian,
title = {{Bayesian Sampling of Structural Ensembles: The Role of Ensemble-Counting Measures}},
author = {Ivan Gilardoni and Giovanni Bussi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.18495},
eprint = {2606.18495},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.18495},
}
RIS
TY - UNPB
TI - Bayesian Sampling of Structural Ensembles: The Role of Ensemble-Counting Measures
AU - Ivan Gilardoni
AU - Giovanni Bussi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2606.18495
UR - https://arxiv.org/abs/2606.18495
AB - Structural ensemble refinement is widely used to integrate molecular simulations with experimental measurements. While most applications focus on the maximum-a-posteriori (MAP) ensemble, Bayesian sampling of the posterior distribution can provide uncertainty estimates and posterior averages for arbitrary observables. A notable step in this direction was introduced by the Bayesian Energy Landscape Tilting (BELT) framework, where sampling is performed on a family of maximum-entropy ensembles parametrized by Lagrange multipliers. Here, we show that Bayesian sampling in this setting requires an explicit choice of ensemble-counting measure. In particular, the flat measure in Lagrange-multiplier space used in the original BELT formulation leads to a posterior distribution that is formally non-normalizable for finite reference trajectories. We propose the Jeffreys measure as an invariant ensemble-counting prescription, restoring normalizability in the finite-sample situations considered here, and providing a consistent definition of posterior averages. Using both an analytically tractable Gaussian model and maximum-entropy refinement of RNA oligomer simulations, we compare different ensemble-counting measures and show that they can significantly affect Bayesian estimates. The resulting methodology has been implemented in the \texttt{MDRefine} software package.
ER -