Reconstructing Free Energy Using Bayesian Thermodynamic Integration

We introduce a new approach for the reconstruction of thermodynamic functions and phase boundaries in two-parameter statistical mechanics systems, without a requirement for knowledge of the system's energy. Our method is based on expressing the Fisher metric in terms of the posterior distributions over a space of external parameters and approximating the metric field by a Hessian of a convex function. We use the proposed approach to reconstruct the partition functions and phase diagrams of the Ising model and the exactly solvable non-equilibrium TASEP without any a priori knowledge about microscopic rules of the models, except for the selection of the range of external parameters.