The problem of low counts in a signal plus noise model

Consider the model X = B + S, where B and S are independent Poisson random variables with means μ and ν, ν is unknown, but μ is known. The model arises in particle physics and some recent articles have suggested conditioning on the observed bound on B; that is, if X = n is observed, then the suggestion is to base inference on the conditional distribution of X given B ≤ n. This conditioning is non-standard in that it does not correspond to a partition of the sample space. It is examined here from the view point of decision theory and shown to lead to admissible formal Bayes procedures.