The Bayesian unlucky broker

We formulate, analyze, and solve a novel topic in detection theory, here referred to as the unlucky broker problem. Suppose you have a standard statistical test between two hypotheses, leading to the optimal Bayesian decision made by exploiting a certain dataset. Later, suppose that part of the data is lost, and we want to remake the test by using the surviving data and the previous decision. What is the best we can do? Such problem, first considered in [1], is faced by standard tools from detection theory. We afford the general form of the optimal detectors, and discuss their operative modalities, emphasizing the intriguing insights hidden in the solution.