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

TL;DR

This paper investigates empirical Bayes methods to quantify the strength of forensic evidence for composite hypotheses, deriving explicit formulas for nonparametric and parametric cases with a fixed prior quantile.

Contribution

It introduces empirical Bayes approaches with a fixed prior quantile to assess evidence strength in forensic settings, including explicit formulas for nonparametric and normal priors.

Findings

Strength of evidence as ratio of suprema of likelihoods.

Derived explicit formulas for nonparametric priors.

Developed strength of evidence functions for normal priors.

Abstract

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.