Fundamental problem of forensic mathematics--the evidential value of a rare haplotype

Y-chromosomal and mitochondrial haplotyping offer special advantages for criminal (and other) identification. For different reasons, each of them is sometimes detectable in a crime stain for which autosomal typing fails. But they also present special problems, including a fundamental mathematical one: When a rare haplotype is shared between suspect and crime scene, how strong is the evidence linking the two? Assume a reference population sample is available which contains n-1 haplotypes. The most interesting situation as well as the most common one is that the crime scene haplotype was never observed in the population sample. The traditional tools of product rule and sample frequency are not useful when there are no components to multiply and the sample frequency is zero. A useful statistic is the fraction κ of the population sample that consists of "singletons" - of once-observed types. A simple argument shows that the probability for a random innocent suspect to match a previously unobserved crime scene type is (1-κ)/n - distinctly less than 1/n, likely ten times less. The robust validity of this model is confirmed by testing it against a range of population models. This paper hinges above all on one key insight: probability is not frequency. The common but erroneous "frequency" approach adopts population frequency as a surrogate for matching probability and attempts the intractable problem of guessing how many instances exist of the specific haplotype at a certain crime. Probability, by contrast, depends by definition only on the available data. Hence if different haplotypes but with the same data occur in two different crimes, although the frequencies are different (and are hopelessly elusive), the matching probabilities are the same, and are not hard to find.