Distance-based and continuum Fano inequalities with applications to statistical estimation

TL;DR

This paper extends Fano inequalities to estimation and continuum settings, providing new lower bounds that simplify proofs of statistical minimax bounds.

Contribution

It introduces two novel extensions of Fano's inequality, one for estimation accuracy and one for continuum volume bounds, advancing theoretical tools in information theory.

Findings

Derived lower bounds for estimation probability within a distance t

Provided volume-based bounds for continuum estimation problems

Simplified proofs of statistical minimax lower bounds

Abstract

In this technical note, we give two extensions of the classical Fano inequality in information theory. The first extends Fano's inequality to the setting of estimation, providing lower bounds on the probability that an estimator of a discrete quantity is within some distance $t$ of the quantity. The second inequality extends our bound to a continuum setting and provides a volume-based bound. We illustrate how these inequalities lead to direct and simple proofs of several statistical minimax lower bounds.