  The issue of discrete probability estimation for samples of small size is
addressed in this study. The maximum likelihood method often suffers
over-fitting when insufficient data is available. Although the Bayesian
approach can avoid over-fitting by using prior distributions, it still has
problems with objective analysis. In response to these drawbacks, a new
theoretical framework based on thermodynamics, where energy and temperature are
introduced, was developed. Entropy and likelihood are placed at the center of
this method. The key principle of inference for probability mass functions is
the minimum free energy, which is shown to unify the two principles of maximum
likelihood and maximum entropy. Our method can robustly estimate probability
functions from small size data.
