Geometry of Discrete Copulas
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Multivariate discrete distributions are fundamental to modeling. Discrete copulas can be used to non-parametrically construct diverse multivariate joint distributions over random variables from estimated univariate marginals. While directly constructing high-dimensional copulas is in general challenging, there is a wealth of techniques for bootstrapping up from estimated bivariate copulas. Thus, it is important to have a variety of methods for selecting bivariate discrete copulas with desirable properties. In this paper, we show that the families of ultramodular discrete copulas and their generalization to convex discrete quasi-copulas admit representations as polytopes, thereby opening the door for computationally efficient techniques from linear optimization for copula selection. In doing so, we draw connections to the prominent Birkhoff and alternating sign matrix polytope in discrete geometry and also generalize some well-known results on these polytopes.
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