The multivariate Dirichlet-multinomial distribution and its application in forensic genetics to adjust for subpopulation effects using the θ-correction

In this paper, we discuss the construction of a multivariate generalisation of the Dirichlet-multinomial distribution. An example from forensic genetics in the statistical analysis of DNA mixtures motivates the study of this multivariate extension. In forensic genetics, adjustment of the match probabilities due to remote ancestry in the population is often done using the so-called θ-correction. This correction increases the probability of observing multiple copies of rare alleles in a subpopulation and thereby reduces the weight of the evidence for rare genotypes.A recent publication by Cowell et al. (2015) showed elegantly how to use Bayesian networks for efficient computations of likelihood ratios in a forensic genetic context. However, their underlying population genetic model assumed independence of alleles, which is not realistic in real populations. We demonstrate how the so-called θ-correction can be incorporated in Bayesian networks to make efficient computations by modifying the Markov structure of Cowell et al. (2015).By numerical examples, we show how the θ-correction incorporated in the multivariate Dirichlet-multinomial distribution affects the weight of evidence.