DNA database matches: A p versus np problem

The evidential value of a unique DNA database match has been extensively discussed. In principle the matter has been mathematically resolved, since the posterior odds on the match being with the trace donor are unambiguously defined. There are multiple ways to express these odds as a product of likelihood ratio and prior odds, and so the mathematics do not immediately tell us what to do in concrete cases, in particular which likelihood ratio to choose for reporting. With p the random match probability for the matching person, if innocent, and n the database size, both 1/p, originating from a suspect-centered framework, and 1/(np), originating from a database-centered framework, arise as likelihood ratio. Both have been defended and both have been criticized in the literature. We will clarify the situation by not introducing models and choices of prior probabilities until they are needed. This allows to derive the posterior odds in their most general form, which applies whenever we know that a single person among a list is not excluded as potential trace donor. We show that we need only three probabilities, that pertain to the observed match, to the database, and to the matching person respectively. How these required probabilities behave in a given context, then, differs from one situation to another. This is understandable since database searches may be done under various circumstances. They may be carried out with or without a suspect already in mind and, depending on the operational procedures, one may or may not be informed about the personal details of the person who gives the match. We show how to evaluate the required probabilities in all such cases. We will motivate why we believe that for some database searches, the 1/p likelihood ratio is more natural, whereas for others, 1/(np) seems the more sensible choice. This is not motivated by the mathematics: mathematically, the approaches are equivalent. It is motivated by considering which model best reflects the actual situation, taking into account what question was asked to begin with, and by the practical consideration of judging which likelihood ratio comes closer to the posterior odds based on the information available in the case. This article is intended to be both a research and a review article, and we end with an in-depth discussion of various arguments that have been brought forward in favor or against either 1/p or 1/(np).