Bayesian Reconstruction of Approximately Periodic Potentials at Finite Temperature

The paper discusses the reconstruction of potentials for quantum systems at finite temperatures from observational data. A nonparametric approach is developed, based on the framework of Bayesian statistics, to solve such inverse problems. Besides the specific model of quantum statistics giving the probability of observational data, a Bayesian approach is essentially based on "a priori" information available for the potential. Different possibilities to implement "a priori" information are discussed in detail, including hyperparameters, hyperfields, and non--Gaussian auxiliary fields. Special emphasis is put on the reconstruction of potentials with approximate periodicity. The feasibility of the approach is demonstrated for a numerical model.