  As a convention, p-value is often computed in frequentist hypothesis testing
and compared with the nominal significance level of 0.05 to determine whether
or not to reject the null hypothesis. The smaller the p-value, the more
significant the statistical test. We consider both one-sided and two-sided
hypotheses in the composite hypothesis setting. For one-sided hypothesis tests,
we establish the equivalence of p-value and the Bayesian posterior probability
of the null hypothesis, which renders p-value an explicit interpretation of how
strong the data support the null. For two-sided hypothesis tests of a point
null, we recast the problem as a combination of two one-sided hypotheses alone
the opposite directions and put forward the notion of a two-sided posterior
probability, which also has an equivalent relationship with the (two-sided)
p-value. Extensive simulation studies are conducted to demonstrate the Bayesian
posterior probability interpretation for the p-value. Contrary to common
criticisms of the use of p-value in evidence-based studies, we justify its
utility and reclaim its importance from the Bayesian perspective, and recommend
the continual use of p-value in hypothesis testing. After all, p-value is not
all that bad.
