Using Bayes factors to limit forensic testimony to forensics: composite hypotheses

In most western legal systems, only the fact-finder (judge or jury) is entrusted to make the ultimate decision in a criminal case. A forensic expert can help the fact-finder by opining on the weight of the forensic evidence given the hypotheses relevant to the case, but is not qualified to give an opinion about the ultimate question(s). When the question is reduced to two simple hypotheses, a Bayes Factor can express the expert's opinion about the extent to which the forensic evidence favours each hypothesis. This paper addresses the situation in which one or both of the hypotheses are composite, that is, embrace more than one possibility. It offers an interval of Bayes Factors, and shows that the proposed interval includes those values, and only those values, of the Bayes Factor supported by possible beliefs of the fact-finder. Shoe prints, tool marks and DNA are discussed in this light if the hypotheses used in the Bayes Factor are composite.