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This convex function penalizes positive densities using negative entropy, with $0\log 0=0$ and infinite value for negative inputs. Its scalar derivative is $\log x$ on $x>0$. On a bounded domain, restricting to uniformly positive bounded functions gives a simple setting for a Fréchet derivative. Its Bregman distance is the generalized Kullback–Leibler divergence. With fixed total mass, minimizing this negative-entropy expression maximizes the corresponding entropy.

For the maximum-entropy regularization functional, differentiation of the scalar integrand gives $(s\log s-s{)}^{\prime }=\log s$. Thus the Fréchet derivative at a positive reference function $y$ is $E^{\prime }(y)=\log y$, and the stated differentiability result gives $\partial E(y)=\{\log y\}$. Consequently,This is the generalized Kullback–Leibler divergence. When both functions integrate to the same mass, the last two terms cancel after integration; for probability density functions it becomes the usual Kullback-Leibler divergence. For two positive functions, adding the reverse distance givesA sufficient setting for the differentiability argument is $L^{\infty }(\Omega )$ with $y$ bounded away from zero. For nonnegative $x$, use $0\log (0/y)=0$ wherever appropriate. Positivity alone in $L^2$ does not automatically provide an open domain on which the Fréchet derivative exists.

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.