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

Rami V. Tabri

TL;DR

The paper develops existence, dual representation, and finite-dimensional approximation results for the $I$-projection in infinite-dimensional moment-inequality models, allowing for both unconditional and conditional constraints and an infinite number of inequalities. It represents the dual variable as a weak vector-valued integral (Gelfand–Pettis integral) and uses a reparametrization to obtain a computable, convex, finite-dimensional approximation scheme based on Sample Average Approximation, with convergence guarantees that accumulation points of discrete optimizers are dual optimizers. The core contributions include an exponential-family form for the $I$-projection density $p_Q = e^{y_0}/\int e^{y_0} dQ$, a weak-star–based dual framework, and practical approximation results that rely on minimal structural assumptions beyond precompactness and boundedness of moment functions. The method is illustrated via unconditional/conditional stochastic dominance constraints and a random-interval (censored data) example, with numerical experiments showing straightforward implementation using off-the-shelf optimization tools. These results enable robust, nonparametric information-projection-based inference and monitoring in settings with complex, infinite-dimensional constraint sets.

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.

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

TL;DR

The paper develops existence, dual representation, and finite-dimensional approximation results for the -projection in infinite-dimensional moment-inequality models, allowing for both unconditional and conditional constraints and an infinite number of inequalities. It represents the dual variable as a weak vector-valued integral (Gelfand–Pettis integral) and uses a reparametrization to obtain a computable, convex, finite-dimensional approximation scheme based on Sample Average Approximation, with convergence guarantees that accumulation points of discrete optimizers are dual optimizers. The core contributions include an exponential-family form for the -projection density , a weak-star–based dual framework, and practical approximation results that rely on minimal structural assumptions beyond precompactness and boundedness of moment functions. The method is illustrated via unconditional/conditional stochastic dominance constraints and a random-interval (censored data) example, with numerical experiments showing straightforward implementation using off-the-shelf optimization tools. These results enable robust, nonparametric information-projection-based inference and monitoring in settings with complex, infinite-dimensional constraint sets.

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 -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.

Paper Structure

This paper contains 14 sections, 10 theorems, 78 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Illustrative Example: Process Monitoring with Interval Censored Data
  3. Setup and Preliminaries
  4. Main Results
  5. Discussion
  6. Theorem \ref{['Thm - Fenchel']}
  7. Approximation of $I$-Projections
  8. Verification of Assumptions
  9. Random Interval Example
  10. $I$-Projection Problem
  11. Numerical Experiments
  12. Intermediate Technical Results
  13. Technical Results for Examples \ref{['Example - SD']} and \ref{['Example - Bhatacharya']}
  14. Proof of Proposition \ref{['Prop - V Random Set']}

Key Result

Theorem 1

Let the constraint set $\mathcal{M}$ be given by (eq - constraint set M), and let the sets $\mathcal{V}$ and $B$ be given by (eq - moment fxns set) and (eq - Binding moment fxns set), respectively. The following statements hold.

Figures (2)

  • Figure 1: Graphs of the approximate $I$-projection, $p_Q$, and its $Q$-density weighted version, $p_Q\,dQ$.
  • Figure 2: Graphs of the approximate $I$-projection, $p_Q$, and its $Q$-density weighted version, $p_Q\,dQ$ based on the modified approximation scheme.

Theorems & Definitions (25)

  • Definition 1: Gelfand-Pettis Integral
  • Theorem 1: Existence and Exponential Family Representation
  • proof
  • Theorem 2: Reparametrization of the dual
  • proof
  • Theorem 3: Discretization of the dual
  • proof
  • Remark 4.1
  • Example 1
  • Example 2
  • ...and 15 more