Bayesian interpretation of discrete class characteristics

Bayesian interpretation of forensic evidence has become dominated by the likelihood ratio (LR) with a large LR generally considered favourable to the prosecution hypothesis, H_P, over the defence hypothesis, H_D. However, the LR simply quantifies by how much the prior odds ratio of the probability of H_P relative to H_D has been improved by the forensic evidence to provide a posterior ratio. Because the prior ratio is mostly neglected, the posterior ratio is largely unknown, regardless of the LR used to improve it. In fact, we show that the posterior ratio will only favour H_P when LR is at least as large as the number of things that could possibly be the source of that evidence, all being equally able to contribute. This restriction severely limits the value of evidence to the prosecution when only a single, discrete class characteristic is used to match a subset of these things to the evidence. The limitation can be overcome by examining more than one individual characteristic, as long as they are independent of each other, as they are for the genotypes at multiple loci combined for DNA evidence. We present a criterion for determining how many such characteristics are required. Finally, we conclude that a frequentist interpretation is inappropriate as a measure of the strength of forensic evidence precisely because it only estimates the denominator of the LR.