Transport, Entropy, and the Cost of Dependence: Maximum Conditional Entropy under Transport Constraints

Maximum-entropy inference under Wasserstein constraints can be computationally challenging due to the underlying optimization over transport couplings. We investigate a tractable alternative based on maximizing conditional entropy under an average transport constraint. On Euclidean spaces, this formulation admits an explicit generalized-Gaussian additive-noise solution, providing a feasible approximation to the Wasserstein-constrained problem without solving an optimal transport problem. We establish quantitative bounds on the resulting optimality gap by showing that the two formulations admit a common feasible-marginal representation, with their discrepancy characterized by the mutual information retained between the nominal and perturbed variables. For discretely supported nominal distributions, we further obtain an exact decomposition of the gap into a marginal-entropy gain and a weighted Jensen-Shannon divergence.