  In the hypothesis testing framework, p-value is often computed to determine
rejection of the null hypothesis or not. On the other hand, Bayesian approaches
typically compute the posterior probability of the null hypothesis to evaluate
its plausibility. We revisit Lindley's paradox (Lindley, 1957) and demystify
the conflicting results between Bayesian and frequentist hypothesis testing
procedures by casting a two-sided hypothesis as a combination of two one-sided
hypotheses along the opposite directions. This can naturally circumvent the
ambiguities of assigning a point mass to the null and choices of using local or
non-local prior distributions. As p-value solely depends on the observed data
without incorporating any prior information, we consider non-informative prior
distributions for fair comparisons with p-value. The equivalence of p-value and
the Bayesian posterior probability of the null hypothesis can be established to
reconcile Lindley's paradox. Extensive simulation studies are conducted with
multivariate normal data and random effects models to examine the relationship
between the p-value and posterior probability.
