| Summary: | Minimum Bayes factors are commonly used to transform two-sided <i>p</i>-values to lower bounds on the posterior probability of the null hypothesis, in particular the bound <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mi>e</mi><mo>·</mo><mi>p</mi><mo>·</mo><mo form="prefix">log</mo><mo>(</mo><mi>p</mi><mo>)</mo></mrow></semantics></math></inline-formula>. This bound is easy to compute and explain; however, it does not behave as a Bayes factor. For example, it does not change with the sample size. This is a very serious defect, particularly for moderate to large sample sizes, which is precisely the situation in which <i>p</i>-values are the most problematic. In this article, we propose adjusting this minimum Bayes factor with the information to approximate an exact Bayes factor, not only when <i>p</i> is a <i>p</i>-value but also when <i>p</i> is a pseudo-<i>p</i>-value. Additionally, we develop a version of the adjustment for linear models using the recent refinement of the Prior-Based BIC.
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