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