# A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation

## TL;DR

This paper introduces a new technique based on binary hypothesis testing to derive tighter, easily computable lower bounds on minimax risk in statistical inference, improving upon traditional Fano's inequality across various applications.

## Contribution

The authors adapt a binary hypothesis testing approach to obtain sharper lower bounds in statistical inference, surpassing Fano's inequality in accuracy and applicability.

## Key findings

- Tighter risk lower bounds in density estimation
- Improved bounds in active learning of classifiers
- Enhanced bounds in compressed sensing

## Abstract

In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically does not give a sharp converse result (error lower bound) for channel coding problems. Moreover, recent work has shown that an argument based on binary hypothesis testing gives tighter results. We adapt this technique to the statistical setting, and argue that Fano's inequality can always be replaced by this approach to obtain tighter lower bounds that can be easily computed and are asymptotically sharp. We illustrate our technique in three applications: density estimation, active learning of a binary classifier, and compressed sensing, obtaining tighter risk lower bounds in each case.