The p-value quantifies the discrepancy between the data and a null hypothesis of interest, usually the assumption of no difference or no effect. A Bayesian approach allows the calibration of p-values by transforming them to direct measures of the evidence against the null hypothesis, so-called Bayes factors. We review the available literature in this area and consider two-sided significance tests for a point null hypothesis in more detail. We distinguish simple from local alternative hypotheses and contrast traditional Bayes factors based on the data with Bayes factors based on p-values or test statistics. A well-known finding is that the minimum Bayes factor, the smallest possible Bayes factor within a certain class of alternative hypotheses, provides less evidence against the null hypothesis than the corresponding p-value might suggest. It is less known that the relationship between p-values and minimum Bayes factors also depends on the sample size and on the dimension of the parameter of interest. We illustrate the transformation of p-values to minimum Bayes factors with two examples from clinical research.

Minimum Bayes factors are commonly used to transform two-sided P values to lower bounds on the posterior probability of the null hypothesis. Several proposals exist in the literature, but none of them depends on the sample size. However, the evidence of a P value against a point null hypothesis is known to depend on the sample size. In this paper we consider P values in the linear model and propose new minimum Bayes factors that depend on sample size and converge to existing bounds as the sample size goes to infinity. It turns out that the maximal evidence of an exact two-1 sided P value increases with decreasing sample size. The effect of adjusting minimum Bayes factors for sample size is shown in two applications.

It is now widely accepted that the standard inferential toolkit used by the scientific research community-null-hypothesis significance testing (NHST)-is not fit for purpose. Yet despite the threat posed to the scientific enterprise, there is no agreement concerning alternative approaches for evidence assessment. This lack of consensus reflects long-standing issues concerning Bayesian methods, the principal alternative to NHST. We report on recent work that builds on an approach to inference put forward over 70 years ago to address the well-known "Problem of Priors" in Bayesian analysis, by reversing the conventional priorlikelihood-posterior ("forward") use of Bayes' theorem. Such Reverse-Bayes analysis allows priors to be deduced from the likelihood by requiring that the posterior achieve a specified level of credibility. We summarise the technical underpinning of this approach, and show how it opens up new approaches to common inferential challenges, such as assessing the credibility of scientific findings, setting them in appropriate context, estimating the probability of successful replications, and extracting more insight from NHST while reducing the risk of misinterpretation. We argue that Reverse-Bayes methods have a key role to play in making Bayesian methods more accessible and attractive for evidence assessment and research synthesis. As a running example we consider a recently published meta-analysis from several randomised controlled trials (RCTs) investigating the association between corticosteroids and mortality in hospitalised patients with COVID-19.

Meta‐analysis provides important insights for evidence‐based medicine by synthesizing evidence from multiple studies which address the same research question. Within the Bayesian framework, meta‐analysis is frequently expressed by a Bayesian normal‐normal hierarchical model (NNHM). Recently, several publications have discussed the choice of the prior distribution for the between‐study heterogeneity in the Bayesian NNHM and used several “vague” priors. However, no approach exists to quantify the informativeness of such priors, and thus, we develop a principled reference analysis framework for the Bayesian NNHM acting at the posterior level. The posterior reference analysis (post‐RA) is based on two posterior benchmarks: one induced by the improper reference prior, which is minimally informative for the data, and the other induced by a highly anticonservative proper prior. This approach applies the Hellinger distance to quantify the informativeness of a heterogeneity prior of interest by comparing the corresponding marginal posteriors with both posterior benchmarks. The post‐RA is implemented in the freely accessible R package ra4bayesmeta and is applied to two medical case studies. Our findings show that anticonservative heterogeneity priors produce platykurtic posteriors compared with the reference posterior, and they produce shorter 95% credible intervals (CrI) and optimistic inference compared with the reference prior. Conservative heterogeneity priors produce leptokurtic posteriors, longer 95% CrI and cautious inference. The novel post‐RA framework could support numerous Bayesian meta‐analyses in many research fields, as it determines how informative a heterogeneity prior is for the actual data as compared with the minimally informative reference prior.