  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.
