  We present methods for inference about relationships between contributors to
a DNA mixture and other individuals of known genotype: a basic example would be
testing whether a contributor to a mixture is the father of a child of known
genotype. The evidence for such a relationship is evaluated as the likelihood
ratio for the specified relationship versus the alternative that there is no
such relationship. We analyse real casework examples from a criminal case and a
disputed paternity case; in both examples part of the evidence was from a DNA
mixture. DNA samples are of varying quality and therefore present challenging
problems in interpretation. Our methods are based on a recent statistical model
for DNA mixtures, in which a Bayesian network (BN) is used as a computational
device; the present work builds on that approach, but makes more explicit use
of the BN in the modelling. The R code for the analyses presented is freely
available as supplementary material.
  We show how additional information of specific genotypes relevant to the
relationship under analysis greatly strengthens the resulting inference. We
find that taking full account of the uncertainty inherent in a DNA mixture can
yield likelihood ratios very close to what one would obtain if we had a single
source DNA profile. Furthermore, the methods can be readily extended to analyse
different scenarios as our methods are not limited to the particular genotyping
kits used in the examples, to the allele frequency databases used, to the
numbers of contributors assumed, to the number of traces analysed
simultaneously, nor to the specific hypotheses tested.
