Decision Theoretic Cutoff and ROC Analysis for Bayesian Optimal Group Testing

TL;DR

This paper develops a Bayesian decision-theoretic framework for group testing, deriving optimal cutoff values and ROC curves to improve defect detection accuracy in large-scale testing scenarios.

Contribution

It introduces a Bayesian optimal approach to group testing, deriving analytical expressions for cutoff values and ROC curves using statistical physics methods.

Findings

Optimal cutoff minimizes expected risk

Analytical ROC and AUC derived for Bayesian setting

Performance matches belief propagation algorithm in large samples

Abstract

We study the inference problem in the group testing to identify defective items from the perspective of the decision theory. We introduce Bayesian inference and consider the Bayesian optimal setting in which the true generative process of the test results is known. We demonstrate the adequacy of the posterior marginal probability in the Bayesian optimal setting as a diagnostic variable based on the area under the curve (AUC). Using the posterior marginal probability, we derive the general expression of the optimal cutoff value that yields the minimum expected risk function. Furthermore, we evaluate the performance of the Bayesian group testing without knowing the true states of the items: defective or non-defective. By introducing an analytical method from statistical physics, we derive the receiver operating characteristics curve, and quantify the corresponding AUC under the Bayesian optimal setting. The obtained analytical results precisely describes the actual performance of the belief propagation algorithm defined for single samples when the number of items is sufficiently large.