Strength of forensic evidence for composite hypotheses: An empirical Bayes view with a fixed prior quantile

Motivated by the forensic problem of determining the strength of evidence of a continuously distributed measurement of evidence, in the situation of composite hypotheses of the prosecutor and the defence concerning a parameter of a parametric model, we consider empirical Bayes methods with a prescribed quantile value for the prior distribution. Firstly we derive the strength of evidence for nonparametric priors. It turns out that we get the by now more or less accepted strength of evidence as the ratio of two suprema, $\sup_{\theta\geq\theta_0}f(x|\theta)/\sup_{\theta<\theta_0}f(x|\theta)$. Here the hypotheses of the prosecutor and defence are given by $H_p: \theta\geq \theta_0$ and $H_d:\theta<\theta_0$. The evidence is seen as a measurement $x$ which is a realization of a random variable with a density $f(x|\theta)$. Secondly we consider a similar parametric empirical Bayes method with a quantile restriction on the prior where the prior distribution is assumed to be normal. Some interesting strength of evidence functions are derived for this situation.