Information Inequalities for Joint Distributions, with Interpretations and Applications

TL;DR

This paper develops generalized entropy inequalities for joint distributions, explores their dualities, and applies these results to combinatorics, matrix theory, and hypothesis testing, revealing interdisciplinary connections.

Contribution

It introduces a broad class of Shannon-type inequalities based on submodular functions, extending classical entropy bounds and providing new applications across multiple fields.

Findings

Derived bounds for joint entropy using subset entropies.

Established a duality between upper and lower entropy bounds.

Applied inequalities to combinatorics, matrix theory, and hypothesis testing.

Abstract

Upper and lower bounds are obtained for the joint entropy of a collection of random variables in terms of an arbitrary collection of subset joint entropies. These inequalities generalize Shannon's chain rule for entropy as well as inequalities of Han, Fujishige and Shearer. A duality between the upper and lower bounds for joint entropy is developed. All of these results are shown to be special cases of general, new results for submodular functions-- thus, the inequalities presented constitute a richly structured class of Shannon-type inequalities. The new inequalities are applied to obtain new results in combinatorics, such as bounds on the number of independent sets in an arbitrary graph and the number of zero-error source-channel codes, as well as new determinantal inequalities in matrix theory. A new inequality for relative entropies is also developed, along with interpretations in terms of hypothesis testing. Finally, revealing connections of the results to literature in economics, computer science, and physics are explored.