Fisher Information and Mutual Information Constraints

TL;DR

This paper explores how Fisher information in statistical models is constrained by mutual information through channel capacity, providing bounds and applications in distributed estimation and data processing inequalities.

Contribution

It establishes a linear scaling bound of Fisher information with mutual information for models with sub-Gaussian score functions and applies this to distributed estimation and data processing inequalities.

Findings

Fisher information scales at most linearly with mutual information.

Derived minimax lower bounds for distributed Gaussian mean estimation.

Established strong data processing inequalities based on mutual information.

Abstract

We consider the processing of statistical samples $X\sim P_\theta$ by a channel $p(y|x)$, and characterize how the statistical information from the samples for estimating the parameter $\theta\in\mathbb{R}^d$ can scale with the mutual information or capacity of the channel. We show that if the statistical model has a sub-Gaussian score function, then the trace of the Fisher information matrix for estimating $\theta$ from $Y$ can scale at most linearly with the mutual information between $X$ and $Y$. We apply this result to obtain minimax lower bounds in distributed statistical estimation problems, and obtain a tight preconstant for Gaussian mean estimation. We then show how our Fisher information bound can also imply mutual information or Jensen-Shannon divergence based distributed strong data processing inequalities.