{"ID":6621264,"CreatedAt":"2026-07-15T01:01:48.440468303Z","UpdatedAt":"2026-07-15T03:28:55.185153975Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.12132","arxiv_id":"2607.12132","title":"Extracting Bayesian Evidence from Frequentist p-Values","abstract":"The $p$-value and the Bayes factor are measures of evidence that are often considered to be philosophically and mathematically incompatible: The $p$-value quantifies conflict between data and $H_0$ (\"surprise\"), whereas the Bayes factor quantifies the relative predictive accuracy of $H_0$ versus $H_1$ (\"evidence\"). We revisit Jeffreys's Approximate Bayes factor (JAB) -- a simple, largely overlooked approximation dating back to the 1930s -- which connects these two paradigms for objective hypothesis testing of the existence of an effect. Under a unit-information prior the approximation requires only the $p$-value and the effective sample size $n_\\text{eff}$. We clarify the core assumptions and boundary conditions for the application of JAB and show across 704 published $t$-tests and 39 comparisons of proportions that JAB approximates objective Bayes factors remarkably well. The connection between $p$-values and JAB has a practical implication: The evidence implied by a $p$-value depends strongly on $n_\\text{eff}$. Conventional verbal labels for $p$-values (e.g., \"strong surprise\" for .001 \u003c $p$ \u003c .01) correspond to similarly graded Bayes factors only around $n_\\text{eff} \\approx 8$; for larger samples the same $p$-value implies weaker evidence. In moderately sized to large samples, $p \u003e .10$ can amount to moderate or even strong evidence for $H_0$. JAB offers a cheap, sample-size-sensitive supplement to $p$-values, computable from routinely reported statistics, that remains valid even under optional stopping.","short_abstract":"The $p$-value and the Bayes factor are measures of evidence that are often considered to be philosophically and mathematically incompatible: The $p$-value quantifies conflict between data and $H_0$ (\"surprise\"), whereas the Bayes factor quantifies the relative predictive accuracy of $H_0$ versus $H_1$ (\"evidence\"). We...","url_abs":"https://arxiv.org/abs/2607.12132","url_pdf":"https://arxiv.org/pdf/2607.12132v1","authors":"[\"Frederik Aust\",\"Samuel Pawel\",\"Eric-Jan Wagenmakers\"]","published":"2026-07-13T20:30:51Z","proceeding":"stat.ME","tasks":"[\"stat.ME\",\"math.ST\"]","methods":"[]","has_code":false}