There has a major problem in the current theory of hypothesis testing in which no unified indicator to evaluate the goodness of various test methods since the cost function or utility function usually relies on the specific application scenario, resulting in no optimal hypothesis testing method. In this paper, the problem of optimal hypothesis testing is investigated based on information theory. We propose an information-theoretic framework of hypothesis testing consisting of five parts: test information (TI) is proposed to evaluate the hypothesis testing, which depends on the a posteriori probability distribution function of hypotheses and independent of specific test methods; precision with the unit of bit is proposed to evaluate the degree of validity of specific test methods; the sampling a posteriori (SAP) probability test method is presented, which makes stochastic selections on the hypotheses according to the a posteriori probability distribution of the hypotheses; the probability of test failure is defined to reflect the probability of the failed decision is made; test theorem is proved that all precision lower than the TI is achievable. Specifically, for every precision lower than TI, there exists a test method with the probability of test failure tending to zero. Conversely, there is no test method whose precision is more than TI. Numerical simulations are performed to demonstrate that the SAP test is asymptotically optimal. In addition, the results show that the precision of the SAP test and the existing test methods, such as the maximum a posteriori probability, expected a posteriori probability, and median a posteriori probability tests, are not more than TI.

翻译：当前假设检验理论存在一个主要问题，即由于代价函数或效用函数通常依赖于具体的应用场景，缺乏统一指标来评估各种检验方法的优劣，导致不存在最优的假设检验方法。本文基于信息论研究最优假设检验问题。我们提出一个包含五部分的信息论假设检验框架：提出检验信息（TI）用于评估假设检验，其依赖于假设的后验概率分布函数，且独立于具体的检验方法；提出以比特为单位的精度指标，用于评估特定检验方法的有效程度；提出采样后验（SAP）概率检验方法，该方法根据假设的后验概率分布对假设进行随机选择；定义检验失败概率以反映做出错误决策的概率；证明检验定理，即所有低于TI的精度均可实现。具体而言，对于每个低于TI的精度，存在一种检验方法使得检验失败概率趋于零。反之，不存在精度超过TI的检验方法。数值仿真表明SAP检验是渐近最优的。此外，结果显示SAP检验以及现有检验方法（如最大后验概率、期望后验概率和中位后验概率检验）的精度均不超过TI。