On maximum entropy and minimum KL-divergence optimization by Gröbner basis methods

In this paper we study constrained maximum entropy and minimum divergence optimization problems, in the cases where integer valued sufficient statistics exists, using tools from computational commutative algebra. We show that the estimation of parametric statistical models in this case can be transformed to solving a system of polynomial equations. We give an implicit description of maximum entropy models by embedding them in algebraic varieties for which we give a Gröbner basis method to compute it. In the cases of minimum KL-divergence models we show that implicitization preserves specialization of prior distribution. This result leads us to a Gröbner basis method to embed minimum KL-divergence models in algebraic varieties.