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

Speaker: Dr Rami Tabri
Affiliation: The University of Sydney
Join via Zoom link: https://uqz.zoom.us/j/82603079317

Abstract 

The Information projection (I-projection) is a relative-entropy minimizer and is core to Information Theory. It arises in many areas of Econometrics and Statistics, such as Constrained Statistical Inference, Variational Bayesian Inference, Computational Optimal Transport, and Large Deviations Theory.

Computational results for I-projections predominantly centre on the finite discrete setup with a finite number of moment equality/inequality constraints. While this setting has useful applications in econometrics and statistics contexts, the finite aspect of that setup excludes many empirically relevant applications involving continuous distributions and allowing for an infinite number of moment inequality restrictions. Constraints of this kind are connected to important applications. For example, the inequality restrictions may define identified sets for parameters under missing data for continuous variables, impose shape constraints on distributions, or define robust orderings of income distributions in terms of poverty and income inequality. Developing the properties of the I-projection in this setting is essential to constructing statistical procedures based on it.

This paper presents new existence, dual representation, and approximation results for the I-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.

About the presenters meeting 

If you would like to meet with Dr Tabri contact: Dr Fu Ouyan.