Reconciling common-source, specific-source, feature-based, and score-based likelihood ratios

We show that the incorporation of any new piece of information allows for improved decision-making in the sense that the expected costs of an optimal decision decrease (or, in boundary cases where no or not enough new information is incorporated, stay the same) whenever this is done by the appropriate update of the probabilities of the hypotheses. Versions of this result have been stated before. However, previous proofs rely on auxiliary constructions with proper scoring rules. We, instead, offer a direct and completely general proof by considering elementary properties of likelihood ratios only. We apply our results to make a contribution to the debates about the use of score-based/feature-based and common/specific-source likelihood ratios. In the literature, these are often presented as different "LR-systems". We argue that the difference between these is simply a matter of which information is processed. There is therefore no such thing as different "LR-systems", there are only differences in the processed information. In particular, despite claims to the contrary, scores can very well be used in forensic practice, and we illustrate this with an extensive example in the DNA kinship context.