Information Processing Equalities and the Information-Risk Bridge

TL;DR

This paper introduces new classes of information measures that unify existing divergences and establish a geometric relationship with Bayes risk, leading to a refined data processing inequality and insights into hypothesis class selection.

Contribution

It develops generalized information measures that encompass many existing divergences and derives a symmetric, geometric relationship with Bayes risk, extending classical inequalities.

Findings

New classes of information measures unify existing divergences.

Established a geometric relationship between information measures and Bayes risk.

Derived a generalized data processing equality that refines classical inequalities.

Abstract

We introduce two new classes of measures of information for statistical experiments which generalise and subsume $\phi$-divergences, integral probability metrics, $\mathfrak{N}$-distances (MMD), and $(f,\Gamma)$ divergences between two or more distributions. This enables us to derive a simple geometrical relationship between measures of information and the Bayes risk of a statistical decision problem, thus extending the variational $\phi$-divergence representation to multiple distributions in an entirely symmetric manner. The new families of divergence are closed under the action of Markov operators which yields an information processing equality which is a refinement and generalisation of the classical data processing inequality. This equality gives insight into the significance of the choice of the hypothesis class in classical risk minimization.