When evaluating DNA evidence within a likelihood ratio framework, should the propositions be exhaustive?

We seek to develop a rational approach to forming propositions when little information is available from the outset, as this often happens in casework. If propositions used when evaluating evidence are not exhaustive (in the context of the case), then there is a theoretical risk that an LR greater than one may be associated with a proposition in the numerator that - if all meaningful propositions had been considered - would in fact have a lower posterior probability after consideration of the evidence. Ideally, all propositions should be considered. However, with multiple propositions, some terms will be larger than others and for simplification very small terms can be neglected without changing the order of magnitude of the value of the evidence (i.e. LR). Our analysis shows that mathematically a contributor's DNA can be assumed to be present under both prosecution and alternative propositions (H_p and H_a) if there is a reasonable prior probability of their DNA being present and their inclusion is supported by the profile. This is because the terms associated to these sub-propositions will dominate our LR. For example, in the absence of specific information, when considering two persons of interest (POI) as potential contributors to a mixed DNA profile we suggest the assumption of one when examining the presence of the other, after checking that both collectively explain the profile well. This represents more meaningful propositions and allows better discrimination. Slooten and Caliebe have shown that the overall LR is the weighted average of LRs with the same number of contributors (NoC) under both propositions. The weights involve both an assessment of the probability of the crime scene DNA profile and the probability of this NoC given the background information.