= Shifted Poisson data fidelity
{title2=$D_g(u)=\int[Tu-g\log(1+Tu)]$}
For $u\ge0$ and a <positivity-preserving operator>, this is the negative <log-likelihood> for independent <Poisson observations> with intensity $1+Tu$, up to constants depending only on the data and the unit background. For bounded $g\ge0$, its scalar <derivatives> are $1-g/(1+s)$ and $g/(1+s)^2$ on $s=Tu\ge0$. It is <convex>, and <strictly convex> in the predicted intensity when $g>0$ almost everywhere. On a unit-area domain, the <Jensen inequality> gives $D_g(u)\ge\|Tu\|_1-\|g\|_\infty\log(1+\|Tu\|_1)$. The shift makes $\log(1+Tu)$ integrable whenever $Tu\in L^1$; no integrability of $\log u$ is needed.
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