Refining Decisions After Losing Data: The Unlucky Broker Problem

Consider a standard statistical hypothesis test, leading to a binary decision made by exploiting a certain dataset. Suppose that, later, part of the data is lost, and we want to refine the test by exploiting both the surviving data and the previous decision. What is the best one can do? Such a question, here referred to as the unlucky broker problem, can be addressed by very standard tools from detection theory, but the solution gives intriguing insights and is by no means obvious. We provide the general form of the optimal detectors and discuss in depth their modus operandi, ranging from simple likelihood ratio tests to more complex behaviors. Limiting cases, where either the surviving data or the initial decision is almost useless, are also discussed.