Likelihood ratios for mixed stains when the number of donors cannot be agreed

Suppose that part of the prosecution's evidence in some crime case is analysis of a blood stain, and that the traits E discovered in the stain suggest multiple donors. Then the prosecution will probably allege some specific inculpatory hypothesis H₀ about the sources of the stain, and P {E | H₀} can be calculated. It is desirable to use this as the numerator of a likelihood ratio. However, in general the obvious denominator P {E | H₀} cannot be calculated, so unless the defense is sufficiently obliging as to stipulate to a specific choice among the potentially infinite number of more or less exculpatory alternative hypotheses, the desired likelihood ratio can't be evaluated. We show that nonetheless, in most cases there is an adequate inequality.