Bitwise MAP estimation for group testing based on holographic transformation

The main contribution of this paper is a non-trivial expression, that is called dual expression, of the posterior values for a non-adaptive group testing problem. The dual expression is useful for exact bitwise MAP estimation. 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. The derivation of the dual expression of posterior values 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.