Signal confidence limits from a neural network data analysis

This paper deals with a situation of some importance for the analysis of experimental data via Neural Network (NN) or similar devices: Let N data be given, such that N = N-s + N-b, where N-s is the number of signals, N-b the number of background events, and both are unknown. Assume that a NN has been trained, such that it will tag signals with efficiency F-s, (0 < F-s < 1) and background data with F-b (0 < F-b < 1). Applying the NN yields N-Y tagged events, We demonstrate that the knowledge of N-Y is sufficient to calculate confidence bounds for the signal likelihood, which have the same statistical interpretation as the Clopper-Pearson bounds for the well-studied case of direct signal observation. Subsequently, we discuss rigorous bounds for the a posteriori distribution function of the signal probability, as well as for the (closely related) likelihood that there are N-s signals in the data. We compare them with results obtained by starting off with a maximum entropy type assumption for the a priori likelihood that there are N-s signals in the data and applying the Bayesian theorem. (C) 1997 Elsevier Science B.V.