The Information Projection in Moment Inequality Models: Existence, Dual Representation, and Approximation

Relative entropy minimization is a widely used method in decisions and operations research that incorporates information through constraints on the underlying probability model. The solution is called information projection, and we present new results for its existence, exponential family representation, and approximation in the infinite-dimensional setting for moment inequality constraint sets, nesting both conditional and unconditional moments and allowing for an infinite number of such inequalities. Our approach considers the Fenchel dual of the relative entropy minimization problem with a key innovation being the exhibition of the dual variable as a weak vector-valued integral, enabling the formulation of a simple approximation scheme. Under suitable assumptions, the values of finite-dimensional convex stochastic programs can approximate the dual problem’s optimum value, and in addition, every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution of the dual problem. We illustrate the verification of assumptions and construction of the approximation scheme’s parameters for the cases of unconditional and conditional first order stochastic dominance constraints and conditions that characterize selectionable distributions for a random set and present numerical experiments demonstrating the simplicity of the approximation scheme.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2024.0568.