The multivariate Dirichlet-multinomial distribution and its application in forensic genetics to adjust for sub-population effects using the {\theta}-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 {\theta}-correction. This correction increases the probability of observing multiple copies of rare alleles and thereby reduces the weight of the evidence for rare genotypes. By numerical examples, we show how the {\theta}-correction incorporated by the use of the multivariate Dirichlet-multinomial distribution affects the weight of evidence. Furthermore, we demonstrate how the {\theta}-correction can be incorporated in a Markov structure needed to make efficient computations in a Bayesian network.