Probability density estimation by regularization method

Learning in neural networks can be considered as estimation of a probability distribution. However it is an ill-posed problem to find the maximum likelihood estimator in the density function space. In this paper, adding a regularization term, a method to select the best density function is proposed. It is shown the regularization method gives a linear sum of Green functions for the best density, whose linear coefficients are given by the solution of a quadratic equation system. Characteristics of the proposed method and differences from Parzen´s method are investigated through computer simulations.