Minimax Robust Detection: Classic Results and Recent Advances

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: ε-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.

Kernel Robust Hypothesis Testing

“…For the Bayesian setting, we compare the performance of our kernel smoothing robust test φ in (20) and the direct robust kernel test φ B in (21). We choose the Gaussian kernel k(x, y) = exp − x−y 2 2σ 2 with the bandwidth parameter σ = 1.…”

“…When the distributions applied in the likelihood ratio test deviate from the true data-generating distributions, the performance of the test may degrade significantly. To address this problem, the approach of robust hypothesis testing is proposed, e.g., [5]- [20], where uncertainty sets are introduced to model the uncertainty in the underlying distributions. Generally, the uncertainty sets are constructed as collections of distributions that lie in the neighborhood of nominal distributions based on some distance measure.…”

“…The robust hypothesis testing problem has been widely studied and various ways of constructing uncertainty sets have been introduced (see, e.g., [7], [20] for a review). The -contamination uncertainty sets and the total variation uncertainty sets were investigated in [5] and a censored likelihood ratio test was constructed and shown to be minimax optimal.…”

Improved Anomaly Detection by Using the Attention-Based Isolation Forest

“…A wide set of machine learning tasks include anomaly detection problems. Therefore, many methods and models have been developed to address them [1,2,3,4,5,6,7,8,9,10,11]. One of the tools for solving the anomaly detection problems is the attention mechanism.…”

“…Due to importance of the anomaly detection problem in many applications, a huge amount of papers covering anomaly detection tasks and studying various aspects of the anomaly detection have been published in the last decades. Many approaches to solving the anomaly detection problem are analyzed in comprehensive survey papers [1,2,3,4,5,6,7,8,9,10,11].…”