  The goal of this paper is to introduce a new theoretical framework for
Optimal Transport (OT), using the terminology and techniques of Fully
Probabilistic Design (FPD). Optimal Transport is the canonical method for
comparing probability measures and has been successfully applied in a wide
range of areas (computer vision Rubner et al. [2004], computer graphics Solomon
et al. [2015], natural language processing Kusner et al. [2015], etc.).
However, we argue that the current OT framework suffers from two shortcomings:
first, it is hard to induce generic constraints and probabilistic knowledge in
the OT problem; second, the current formalism does not address the question of
uncertainty in the marginals, lacking therefore the mechanisms to design robust
solutions. By viewing the OT problem as the optimal design of a probability
density function with marginal constraints, we prove that OT is an instance of
the more generic FPD framework. In this new setting, we can furnish the OT
framework with the necessary mechanisms for processing probabilistic
constraints and deriving uncertainty quantifiers, hence establishing a new
extended framework, called FPD-OT. Our main contribution in this paper is to
establish the connection between OT and FPD, providing new theoretical insights
for both. This will lay the foundations for the application of FPD-OT in a
subsequent work, notably in processing more sophisticated knowledge
constraints, as well as in designing robust solutions in the case of uncertain
marginals.
