The p-value interpreted as the posterior probability of explaining the data: Applications to multiple testing and to restricted parameter spaces

Failures to replicate the results of scientific studies are often attributed to misinterpretations of the p-value. The p-value may be interpreted as an approximate posterior probability, not that the null hypothesis is true but rather that it explains the data as well as the data-generating distribution. That posterior probability modifies the p-value in the following two broad areas of application, leading to new methods of hypothesis testing and effect size estimation. First, when corrected for multiple comparisons, the posterior probability that the null hypothesis adequately explains the data overcomes both the conservative bias of corrected p-values and the anti-conservative bias of commonly used false discovery rate methods. Second, the posterior probability that the null hypothesis adequately explains the data, conditional on a parameter restriction, transforms the p-value in such a way as to overcome difficulties in restricted parameter spaces.