On Multiple Hypothesis Testing with Rejection Option

TL;DR

This paper investigates the tradeoffs and optimal strategies in multiple hypothesis testing with a rejection option, focusing on discrete sources and providing geometric insights into decision schemes.

Contribution

It introduces a framework for analyzing error exponents in multiple hypothesis testing with rejection, including specialized results for discrete memoryless sources and geometric interpretations.

Findings

Optimal decision strategies minimize error probabilities.

Rejection decisions can have lower error than choosing among hypotheses.

Geometric interpretations aid understanding of bounds in hypothesis testing.

Abstract

We study the problem of multiple hypothesis testing (HT) in view of a rejection option. That model of HT has many different applications. Errors in testing of M hypotheses regarding the source distribution with an option of rejecting all those hypotheses are considered. The source is discrete and arbitrarily varying (AVS). The tradeoffs among error probability exponents/reliabilities associated with false acceptance of rejection decision and false rejection of true distribution are investigated and the optimal decision strategies are outlined. The main result is specialized for discrete memoryless sources (DMS) and studied further. An interesting insight that the analysis implies is the phenomenon (comprehensible in terms of supervised/unsupervised learning) that in optimal discrimination within M hypothetical distributions one permits always lower error than in deciding to decline the set of hypotheses. Geometric interpretations of the optimal decision schemes are given for the current and known bounds in multi-HT for AVS's.