On the irreconcilability of frequentist and Bayesian techniques in point null hypothesis testing

Resumen

A Bayesian test for H0 : µ = µ0 versus H1 : µ 6= µ0, with a given prior density ¼(µ), is developed. The methodology consists of ¯xing a sphere of radius ± around µ0 and assigning to H0 a prior mass, ¼0, computed by integrating the density ¼(µ) over B (µ0; ±) and spreading the remainder, 1 ¡ ¼0, over H1 according to ¼(µ). The ultimate goal of this paper is to show when p-values and posterior probabilities can give rise to the same decision in the following sense. For a ¯xed p¤ 2 (0; 1), when there exist `1 · `2 such that for any observed data, a Bayesian with ¼0 2 (`1; `2) would reach the same conclusion as a frequentist who uses p¤ as the level of signi¯cance?. A theorem that provides the required constructions of `1 and `2 under the veri¯cation of a su±cient condition (`1 · `2) is proved. The innovation with respect to other works about comparing frequentist and Bayesian approaches in testing problem consists of investigating when the same decision is reached with both methods instead of comparing numerically the p-value with the posterior probability or the Bayes factor. Lindley's paradox and other well known examples are reviewed.