On the existence and characterization of the maxent distribution under general moment inequality constraints

TL;DR

This paper establishes broad conditions for the existence and characterization of maximum entropy distributions under general moment inequality constraints, extending prior work to unbounded supports and providing analytical insights.

Contribution

It introduces new sufficient conditions for maxent distribution existence under general inequalities and derives an analytical characterization using infinite-dimensional optimization theory.

Findings

Conditions for maxent existence with unbounded support

Analytical form of maxent distribution derived

Properties of convex coercive functions discussed

Abstract

A broad set of sufficient conditions that guarantees the existence of the maximum entropy (maxent) distribution consistent with specified bounds on certain generalized moments is derived. Most results in the literature are either focused on the minimum cross-entropy distribution or apply only to distributions with a bounded-volume support or address only equality constraints. The results of this work hold for general moment inequality constraints for probability distributions with possibly unbounded support, and the technical conditions are explicitly on the underlying generalized moment functions. An analytical characterization of the maxent distribution is also derived using results from the theory of constrained optimization in infinite-dimensional normed linear spaces. Several auxiliary results of independent interest pertaining to certain properties of convex coercive functions are also presented.