 Poisson data fidelity

For positive observations $y$ and positive predicted intensities $s$, this is the generalized Kullback–Leibler divergence with observation in the first argument. Its scalar derivative in $s$ is $1-y/s$ and its second derivative is $y/s^2>0$. It differs from the divergence of a reconstructed density from a reference density by which argument is varied. A linear prediction $s=Kx$ gives derivative $K^{\ast }(1-y/(Kx))$ through the adjoint operator, on suitable admissible function spaces. This fidelity arises from the negative log likelihood of a Poisson observation model, after removal of terms depending only on observations.