论文标题
高维Bernoulli分布：代数表示和应用
High dimensional Bernoulli distributions: algebraic representation and applications
论文作者
论文摘要
本文的主要贡献是找到多变量Bernoulli分布的类$ \ Mathcal {f} _d（p）$的表示形式，其平均$ p $使我们可以在任何维度上分析地找到其生成器。我们将$ \ Mathcal {f} _d（p）$映射到一个值的理想，我们证明可以从有限的简单多项式集合中生成类$ \ mathcal {f} _d（p）$。我们提出两个应用程序。首先，我们表明多项式发电机有助于找到凸多属$ \ Mathcal {f} _d（p）$的极端点。其次，我们解决了在凸顺序中确定下限的问题，以确定具有给定边缘的多元伯努利分布的总和，但具有未指定的依赖性结构。
The main contribution of this paper is to find a representation of the class $\mathcal{F}_d(p)$ of multivariate Bernoulli distributions with the same mean $p$ that allows us to find its generators analytically in any dimension. We map $\mathcal{F}_d(p)$ to an ideal of points and we prove that the class $\mathcal{F}_d(p)$ 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 $\mathcal{F}_d(p)$ 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.
