  When the infection prevalence of a disease is low, Dorfman showed 80 years
ago that testing groups of people can prove more efficient than testing people
individually. Our goal in this paper is to propose new group testing algorithms
that can operate in a noisy setting (tests can be mistaken) to decide
adaptively (looking at past results) which groups to test next, with the goal
to converge to a good detection, as quickly, and with as few tests as possible.
We cast this problem as a Bayesian sequential experimental design problem.
Using the posterior distribution of infection status vectors for $n$ patients,
given observed tests carried out so far, we seek to form groups that have a
maximal utility. We consider utilities such as mutual information, but also
quantities that have a more direct relevance to testing, such as the AUC of the
ROC curve of the test. Practically, the posterior distributions on $\{0,1\}^n$
are approximated by sequential Monte Carlo (SMC) samplers and the utility
maximized by a greedy optimizer. Our procedures show in simulations significant
improvements over both adaptive and non-adaptive baselines, and are far more
efficient than individual tests when disease prevalence is low. Additionally,
we show empirically that loopy belief propagation (LBP), widely regarded as the
SoTA decoder to decide whether an individual is infected or not given previous
tests, can be unreliable and exhibit oscillatory behavior. Our SMC decoder is
more reliable, and can improve the performance of other group testing
algorithms.
