Quasi-continuous maximum entropy distribution approximation with kernel density

This paper extends maximum entropy estimation of discrete probability distributions to the continuous case. This transition leads to a nonparametric estimation of a probability density function, preserving the maximum entropy principle. Furthermore, the derived density estimate provides a minimum mean integrated square error. In a second step it is shown, how boundary conditions can be included, resulting in a probability density function obeying maximum entropy. The criterion for deviation from a reference distribution is the Kullback-Leibler Entropy. It is further shown, how the characteristics of a particular distribution can be preserved by using integration kernels with mimetic properties.