  A Bayesian approach is developed to determine quantum mechanical potentials
from empirical data. Bayesian methods, combining empirical measurements and "a
priori" information, provide flexible tools for such empirical learning
problems. The paper presents the basic theory, concentrating in particular on
measurements of particle coordinates in quantum mechanical systems at finite
temperature. The computational feasibility of the approach is demonstrated by
numerical case studies. Finally, it is shown how the approach can be
generalized to such many-body and few-body systems for which a mean field
description is appropriate. This is done by means of a Bayesian inverse
Hartree-Fock approximation.
