A Bayesian approach is developed to determine quantum-mechanical potentials
from empirical data. Bayesian methods, combining empirical measurements an
d 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 fi
nite temperature. The computational feasibility of the approach is demonstr
ated by numerical case studies. Finally, it is shown how the approach can b
e 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 Ha
rtree-Fock approximation.