Negative conclusion cases: a proposal for likelihood ratio evaluation

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Abstract
The use of Bayesian approach in forensic science requires the evaluation of the likelihood ratio related to the crime scene evidence event E and the suspect characteristic event C. This evaluation fails when the two events are disjoint, i.e. the evidence E is not compatible with the characteristic C of the suspect, and then the case-work has a negative conclusion. This situation is very common, especially using continuous variables, e.g. height, refractive index and voice frequencies. In particular, in standard approach there is no difference between an evidence E close to C (for instance, heights with 1 cm of difference) and an evidence far from C. We propose a method of calculation of the likelihood ratio, based on a bivariate representation of the database, supposed to be Gaussian, with a correlation coefficient r > 0. The likelihood ratio, calculated with this method, 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. Application in the case of height is performed using Italian Carabinieri database.

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