Existence of the signal in the signal plus background model

Searching for evidence of neutrino oscillations is an important problem in particle physics. Suppose that evidence for neutrino oscillations from an LSND experiment reports a significant positive oscillation probability, but that the LSND result is not confirmed by other experiments. In statistics, such a problem can be proposed as the detection of signal events in the Poisson signal plus background model. Suppose that an observed count $X$ is of the form $X=B+S$, where the background $B$ and the signal $S$ are independent Poisson random variables with parameters $b$ and $\theta$ respectively, $b$ is known but $\theta$ is not. Some recent articles have suggested conditioning on the observed bound for $B$; that is, if $X=n$ is observed, the suggestion is to base the inference on the conditional distribution of $X$ given $B\le n$. This suggestion is used here to derive an estimator of the probability of the existence of the signal event. The estimator is examined from the view of decision theory and is shown to be admissible.