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Free energies as a function of a selected set of collective variables arecommonly computed in molecular simulation and of significant value inunderstanding and engineering molecular behavior. These free energy surfacesare most commonly estimated using variants of histogramming techniques, butsuch approaches obscure two important facets of these functions. First, theempirical observations along the collective variable are defined by an ensembleof discrete observations and the coarsening of these observations into ahistogram bins incurs unnecessary loss of information. Second, the free energysurface is itself almost always a continuous function, and its representationby a histogram introduces inherent approximations due to the discretization. Inthis study, we relate the observed discrete observations from biasedsimulations to the inferred underlying continuous probability distribution overthe collective variables and derive histogram-free techniques for estimatingthis free energy surface. We reformulate free energy surface estimation asminimization of a Kullback-Leibler divergence between a continuous trialfunction and the discrete empirical distribution and show that this isequivalent to likelihood maximization of a trial function given a set ofsampled data. We then present a fully Bayesian treatment of this formalism,which enables the incorporation of powerful Bayesian tools such as theinclusion of regularizing priors, uncertainty quantification, and modelselection techniques. We demonstrate this new formalism in the analysis ofumbrella sampling simulations for the $\chi$ torsion of a valine sidechain inthe L99A mutant of T4 lysozyme with benzene bound in the cavity.

Statistically optimal continuous free energy surfaces from biased simulations and multistate reweighting