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

This paper presents new existence, dual representation, and approximation results for the information projection (I-projection) in the infinite-dimensional setting for moment inequality constraint sets. These results are established under a general specification of the moment inequalities, nesting both conditional and unconditional moments 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, enabling the formulation of an approximation scheme of the I-projection's equivalent Fenchel dual problem. In particular, we show under suitable assumptions that the values of finite-dimensional programs can approximate the dual problem's optimum value and that, in addition, every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution of the dual problem.