The problem of signal detection under an unknown background can be framed as one of inferring the weight of a mixture model with one misspecified component. Banerjee and Algeri (2026) show that, for this problem, the conservativeness of the inference is entirely determined by one single parameter, called the compensator. They demonstrate that, when the data are independent and identically distributed, an inferential approach based on the compensator circumvents the need to estimate the density of the misspecified component and the associated challenges. The main purpose of this manuscript is to broaden the scope of such an approach and extend it to the case in which, as is often encountered in modern experiments in physics and astronomy, the data consist of Poisson counts observed over a large number of bins.

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