  Copulas are a powerful tool to model dependence between the components of a
random vector. One well-known class of copulas when working in two dimensions
is the Farlie-GumbelMorgenstern (FGM) copula since their simple analytic shape
enables closed-form solutions to many problems in applied probability. However,
the classical definition of high-dimensional FGM copula does not enable a
straightforward understanding of the effect of the copula parameters on the
dependence, nor a geometric understanding of their admissible range. We
circumvent this issue by studying the FGM copula from a probabilistic approach
based on multivariate Bernoulli distributions. This paper studies
high-dimensional exchangeable FGM copulas, a subclass of FGM copulas. We show
that dependence parameters of exchangeable FGM can be expressed as convex hulls
of a finite number of extreme points and establish partial orders for different
exchangeable FGM copulas (including maximal and minimal dependence). We also
leverage the probabilistic interpretation to develop efficient sampling and
estimating procedures and provide a simulation study. Throughout, we discover
geometric interpretations of the copula parameters that assist one in decoding
the dependence of high-dimensional exchangeable FGM copulas.
