High dimensional Bernoulli distributions: algebraic representation and applications

The main contribution of this paper is to find a representation of the class Fd(p) of multi- variate Bernoulli distributions with the same mean p that allows us to find its generators analytically in any dimension. We map Fd(p) to an ideal of points and we prove that the class Fd(p) can be generated from a finite set of simple polynomials. We present two applications. Firstly, we show that polynomial generators help to find extremal points of the convex polytope Fd(p) in high dimensions. Secondly, we solve the problem of determining the lower bounds in the convex order for sums of multivariate Bernoulli distributions with given margins, but with an unspecified dependence structure.