  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.
