Choquet-Wasserstein pseudo-distances via optimal transport under partially specified marginal probabilities

We focus on the marginal problem by relaxing the requirement of completely specified marginal probabilities, and referring to Dempster-Shafer theory to encode such partial probabilistic information. We investigate the structure of a suitable set of bivariate joint belief functions having fixed marginals by relying on copula theory. The chosen set of joint belief functions is used to minimize a functional of a given cost function, so as to select an optimal imprecise transport plan in the form of a joint belief function. We formulate two Kantorovich-like optimal transport problems by seeking to minimize the Choquet integral of the cost function with respect to either the reference set of joint belief functions or their dual plausibility functions. We give a noticeable application by choosing a metric as cost function: this permits to define pessimistic and optimistic Choquet-Wasserstein pseudo-distances, that can be used to compare belief functions on the same space. We finally deal with the problem of approximating a belief function with an element of a distinguished class of belief functions, by minimizing one of the two Choquet-Wasserstein pseudo-distances.