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We consider the problem of group testing with sum observations and noiselessanswers, in which we aim to locate multiple objects by querying the number ofobjects in each of a sequence of chosen sets. We study a probabilistic settingwith entropy loss, in which we assume a joint Bayesian prior density on thelocations of the objects and seek to choose the sets queried to minimize theexpected entropy of the Bayesian posterior distribution after a fixed number ofquestions. We present a new non-adaptive policy, called the dyadic policy, showit is optimal among non-adaptive policies, and is within a factor of two ofoptimal among adaptive policies. This policy is quick to compute, itsnonadaptive nature makes it easy to parallelize, and our bounds show itperforms well even when compared with adaptive policies. We also study anadaptive greedy policy, which maximizes the one-step expected reduction inentropy, and show that it performs at least as well as the dyadic policy,offering greater query efficiency but reduced parallelism. Numericalexperiments demonstrate that both procedures outperform a divide-and-conquerbenchmark policy from the literature, called sequential bifurcation, and showhow these procedures may be applied in a stylized computer vision problem.

Probabilistic Group Testing under Sum Observations: A Parallelizable 2-Approximation for Entropy Loss