High dimensional Bernoulli distributions: algebraic representation and applications

2022/05/25 by Roberto Fontana, Fontana, Roberto, Patrizia Semeraro +1 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Mathematical functions and polynomials #Polynomial and algebraic computation #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2205.12744

openalex publication_date 2022/05/25 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28

Abstract

The main contribution of this paper is to find a representation of the class F_d(p) of multivariate Bernoulli distributions with the same mean p that allows us to find its generators analytically in any dimension. We map F_d(p) to an ideal of points and we prove that the class F_d(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 F_d(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.

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