Negative conclusion cases: further proposal for likelihood ratio evaluation

摘要

The use of Bayesian approach in forensic science requires the evaluation of likelihood ratio related to the crime scene evidence (denoted as event E) and the suspect characteristic (denoted as event C). This evaluation is trivially naught when the two events are disjoint, and it is a fraction with zero denominator when E ⊆ C. In this paper, we define an extended likelihood ratio, which is well-defined and different from zero for any pair E and C, using the theory of copulas. This theory allows us to extend our previous paper (Cardinetti & Cammarota, 2005), that was restricted to Gaussian database, to a general database. For different kinds of copulas (Fréchet, Cuadras–Augé and normal copulas), with a correlation coefficient r (with 0 < r < 1), we show that the likelihood ratio has larger values when both the events are in the tail of the distribution, as expected. Moreover, it reduces to the standard one when r tends to 1, and its value is 1 in the case of independence (r = 0). We propose three different approaches in choosing the parameter r in real cases. In the first, r is chosen as a fixed parameter (for instance r = 0.95); in the second, in order to overestimate the extended likelihood ratio LRr, the value of the parameter r should be that which corresponds to the supremum of LRr. In the third approach, a choice of a maximal score K for the likelihood ratio should determine the value of the parameter r. Application in the case of height is performed using Italian Carabinieri database.