Statistical analysis of DNA mixtures for forensic identification is known to pose computational challenges due to the enormous state space of possible DNA profiles. We describe a general method for computing the expectation of a product of discrete random variables using auxiliary variables and probability propagation in a Bayesian network. We propose a Bayesian network representation for genotypes, allowing computations to be performed locally involving only a few alleles at each step. Exploiting appropriate auxiliary variables in combination with this representation allows efficient computation of the likelihood function and prediction of genotypes of unknown contributors. Importantly, we exploit the computational structure to introduce a novel set of diagnostic tools for assessing the adequacy of the model for describing a particular dataset.