Comment on: Why the effect of prior odds should accompany the likelihood ratio when reporting DNA evidence

Abstract

Meester and Sjerps tackle difficult issues, and many of their comments and suggestions have much to commend them. However, I do not think that they have resolved the difficulties fully satisfactorily. I first part company with them when they speak, in the third paragraph of the Introduction, of a 'randomly selected member of the population', with many subsequent repetitions of similar phrases. In crime investigations no-one, we hope, is selected randomly. The concept of 'random man' has captured the imagination of statisticians and the general public since Quetelet in the mid-19th century, in some instances to good effect. In the forensic identification setting, however, I believe that the concept is unnecessary and dangerous. Consider perhaps the simplest setting of a single individual, the suspect S, whose DNA profile matches that obtained from a single crime stain and is presumed to be that of the culprit C. Here, the relevant version of Bayes' theorem is P(C=S|E) = P(E|C=S)P(C=S) P(E|C=S)P(C=S) + P(E|C =S)P(C =S) (1) where E denotes the DNA evidence; all the probabilities are implicitly conditional on any other evidence and background information. The denominator of (1) cannot readily be evaluated in this form because the alternative hypothesis C = S is a compound of many possible alternatives that specify various probabilities for E. The most natural way to partition C = S into useful sub-hypotheses is to consider all the alternative possible culprits: P(E|C =S)P(C =S) = X P(E|C=X)P(C=X), (2) where the summation is over all individuals X who might be C, excluding S. Substituting (2) into (1), and writing L X = P(E|C=X) P(E|C=S) for the likelihood ratio comparing the hypotheses that X and S, respectively, is the culprit, and π X = P(C=X) P(C=S) , †