# An Introductory Guide to Fano's Inequality with Applications in Statistical Estimation

## TL;DR

This paper surveys Fano's inequality and its variants, highlighting their importance in establishing impossibility results across various statistical estimation problems in information theory.

## Contribution

It provides a comprehensive framework and key tools for applying Fano's inequality to diverse statistical estimation challenges.

## Key findings

- Fano's inequality is versatile for impossibility proofs.
- Applications include group testing, graphical models, and sparse regression.
- The survey covers techniques and examples across multiple domains.

## Abstract

Information theory plays an indispensable role in the development of algorithm-independent impossibility results, both for communication problems and for seemingly distinct areas such as statistics and machine learning. While numerous information-theoretic tools have been proposed for this purpose, the oldest one remains arguably the most versatile and widespread: Fano's inequality. In this chapter, we provide a survey of Fano's inequality and its variants in the context of statistical estimation, adopting a versatile framework that covers a wide range of specific problems. We present a variety of key tools and techniques used for establishing impossibility results via this approach, and provide representative examples covering group testing, graphical model selection, sparse linear regression, density estimation, and convex optimization.