Optimizing nonadaptive group tests for objects with heterogeneous priors

We investigate nonadaptive group testing designs for heterogeneous mixtures of objects, independently positive with individual prior probabilities. In our model of the prior probabilities, the objects occur in one of several disjoint subsets and the number of positives in each subset is known. Furthermore, the positives are "uniformly distributed" within the subsets. The expected number of unresolved negative objects is minimized, and a unique global minimum is found for a family of stochastic, random incidence designs: all v group tests are constructed independently. The optimum incidence probabilities for the objects are well approximated by an asymptotic power series in v(-1). We find the three leading coefficients of this series. The dependence of the optimum incidence probability upon the prior probability is, to leading order, logarithmic. Objects with larger prior probability of being positive have smaller optimum incidence probability. Furthermore, this logarithmic dependence can be nonnegligible for screening collections of cloned DNA sequences.