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

TL;DR

This paper develops new theoretical results for the existence, dual representation, and approximation of the information projection in infinite-dimensional moment inequality models, with practical numerical demonstrations.

Contribution

It introduces a novel approximation scheme for the dual problem of the information projection, applicable to complex moment inequality models including conditional and unconditional cases.

Findings

Finite-dimensional approximations converge to the dual problem's optimum

Optimal solutions of approximations approach the true dual solution

Numerical experiments confirm the scheme's simplicity and effectiveness

Abstract

This paper presents new existence, dual representation, and approximation results for the information projection in the infinite-dimensional setting for moment inequality models. These results are established under a general specification of the moment inequality model, nesting both conditional and unconditional models, and allowing for an infinite number of such inequalities. An essential innovation of the paper is the exhibition of the dual variable as a weak vector-valued integral to formulate an approximation scheme of the $I$-projection's equivalent Fenchel dual problem. In particular, it is shown under suitable assumptions that the dual problem's optimum value can be approximated by the values of finite-dimensional programs and that, in addition, every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution for the dual problem. This paper illustrates the verification of assumptions and the construction of the approximation scheme's parameters for the cases of unconditional and conditional first-order stochastic dominance constraints and dominance conditions that characterize selectionable distributions for a random set. The paper also includes numerical experiments based on these examples that demonstrate the simplicity of the approximation scheme in practice and its straightforward implementation using off-the-shelf optimization methods.