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.
翻译:我们从决定理论的角度研究小组测试中的推论问题,以辨别有缺陷的物品。我们引入了贝叶斯推论,并考虑了了解测试结果真实基因化过程的巴伊西亚最佳环境;我们根据曲线(AUC)下的区域,以巴伊西亚最佳环境的后边概率作为诊断变量,证明了巴伊西亚最佳环境的后边概率是否充分。我们利用后边概率,得出得出产生最低预期风险功能的最佳截断值的一般表现。此外,我们评估了巴伊西亚群体测试的性能,而不知道项目的真实状况:缺陷或非缺陷。我们从统计物理学中引入了分析方法,我们从巴伊西亚最佳环境中得出了接收器操作特征曲线,并对相应的ACU进行了量化。分析结果准确地描述了在物品数量足够大的情况下为单一样品确定的信仰传播算法的实际表现。