  We describe different Bayesian ensemble refinement methods, examine their
interrelation, and discuss their practical application. With ensemble
refinement, the properties of dynamic and partially disordered (bio)molecular
structures can be characterized by integrating a wide range of experimental
data, including measurements of ensemble-averaged observables. We start from a
Bayesian formulation in which the posterior is a functional that ranks
different configuration space distributions. By maximizing this posterior, we
derive an optimal Bayesian ensemble distribution. For discrete configurations,
this optimal distribution is identical to that obtained by the maximum entropy
"ensemble refinement of SAXS" (EROS) formulation. Bayesian replica ensemble
refinement enhances the sampling of relevant configurations by imposing
restraints on averages of observables in coupled replica molecular dynamics
simulations. We show that the strength of the restraint should scale linearly
with the number of replicas to ensure convergence to the optimal Bayesian
result in the limit of infinitely many replicas. In the "Bayesian inference of
ensembles" (BioEn) method, we combine the replica and EROS approaches to
accelerate the convergence. An adaptive algorithm can be used to sample
directly from the optimal ensemble, without replicas. We discuss the
incorporation of single-molecule measurements and dynamic observables such as
relaxation parameters. The theoretical analysis of different Bayesian ensemble
refinement approaches provides a basis for practical applications and a
starting point for further investigations.
