  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;
accuracy 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 accuracy lower than the TI is achievable.
Specifically, for every accuracy lower than TI, there exists a test method with
the probability of test failure tending to zero. Conversely, there is no test
method whose accuracy is more than TI. Numerical simulations are performed to
demonstrate that the SAP test is asymptotically optimal. In addition, the
results show that the accuracy 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.
