In this example we study the classic "on-off" problem [14] of simultaneous measurements of two Poisson processes: p(x 1 , x 2 |µ, ν) = Pois(x 1 |µs + νb)Pois(
x 2 |ντ b),(10)
where the hyperparameters s, b indicate the nominally expected signal and background counts. The hyperparameter τ describes the relationship in measurement time between the two processes. The signal strength µ is the parameter of interest while the scaling factor for the background ν is a nuisance parameter. This model deviates from a pure Gaussian setup and introduces a more complicated relationship between **data** and parameters. The results are shown in Figure 2. With hyperparameters set at s = 15, b = 70, τ = 1 the model is comfortably in the asymptotic regime but with τ ∼ 1 significant deviation the profiled values of the nuisance parameters is expected as one moves away of the maximum-likelihood estimate. As in the Gaussian case, the neural network directly approximates the profile likelihood ratio to very high degree of precision.