An Introductory Guide to Fano's Inequality with Applications in Statistical Estimation
It provides a guide for researchers in statistics and machine learning to apply information-theoretic methods for impossibility proofs, but it is incremental as a survey.
The paper surveys Fano's inequality and its variants as tools for establishing algorithm-independent impossibility results in statistical estimation, covering applications like group testing and sparse linear regression.
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.