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.