Bitwise MAP Algorithm for Group Testing based on Holographic Transformation

In this paper, an exact bitwise MAP (Maximum A Posteriori) estimation algorithm for group testing problems is presented. We assume a simplest non-adaptive group testing scenario including N -objects with binary status and M disjunctive tests. If a group contains a positive object, the test result for the group is assumed to be one; otherwise, the test result becomes zero. Our inference problem is to evaluate the posterior probabilities of the objects from the observation of M test results and from our knowledge on the prior probabilities for objects. If the size of each group is bounded by a constant, a naive inference algorithm requires O(N2 )-time for computing the posterior probabilities for objects. Our algorithm runs with O(N2 )-time, which is exponentially faster than the naive inference algorithm under a common situation with M << N . The heart of the algorithm is the dual expression of the posterior values. The derivation of the dual expression can be naturally described based on a holographic transformation to the normal factor graph (NFG) representing the inference problem. In order to handle OR constraints in the NFG, we introduce a novel holographic transformation that converts an OR function to a function similar to an EQUAL function.