Minimax Robust Detection: Classic Results and Recent Advances

TL;DR

This paper reviews minimax robust hypothesis testing, covering classical results, recent advances, and practical applications, with a focus on binary and multiple hypotheses, uncertainty modeling, and sequential detection techniques.

Contribution

It provides a comprehensive overview of minimax robust detection, including explicit formulas for least favorable distributions and new insights into multi-hypothesis testing and practical applications.

Findings

Explicit expressions for least favorable distributions under various uncertainty models

Sequential detection enables minimax optimal tests for multiple hypotheses

Robust detectors are effective in practical scenarios like ground penetrating radar

Abstract

This paper provides an overview of results and concepts in minimax robust hypothesis testing for two and multiple hypotheses. It starts with an introduction to the subject, highlighting its connection to other areas of robust statistics and giving a brief recount of the most prominent developments. Subsequently, the minimax principle is introduced and its strengths and limitations are discussed. The first part of the paper focuses on the two-hypothesis case. After briefly reviewing the basics of statistical hypothesis testing, uncertainty sets are introduced as a generic way of modeling distributional uncertainty. The design of minimax detectors is then shown to reduce to the problem of determining a pair of least favorable distributions, and different criteria for their characterization are discussed. Explicit expressions are given for least favorable distributions under three types of uncertainty: $\varepsilon$-contamination, probability density bands, and $f$-divergence balls. Using examples, it is shown how the properties of these least favorable distributions translate to properties of the corresponding minimax detectors. The second part of the paper deals with the problem of robustly testing multiple hypotheses, starting with a discussion of why this is fundamentally different from the binary problem. Sequential detection is then introduced as a technique that enables the design of strictly minimax optimal tests in the multi-hypothesis case. Finally, the usefulness of robust detectors in practice is showcased using the example of ground penetrating radar. The paper concludes with an outlook on robust detection beyond the minimax principle and a brief summary of the presented material.