Bayes estimator under weighted absolute loss ($a\in \mathrm{Median}(w(\theta )\pi (\theta \mid x))$)

For loss $L(\theta ,a)=w(\theta )|\theta -a|$, assume $w\ge 0$, $0<\mathbb{E}[w(\Theta )\mid x]<\infty$ and $\mathbb{E}[w(\Theta )|\Theta |\mid x]<\infty$. The posterior expected loss is a positive constant times absolute-error loss under the posterior distribution reweighted by $w$. Thus its minimizers are the medians of that weighted posterior distribution. More precisely, the weighted cumulative distribution function $F_w$ satisfies $F_w(a^{-})\le 1/2\le F_w(a)$; the one-sided slopes of the convex loss give this condition.

For relative absolute loss on positive parameters, $w(\theta )=1/\theta$. With a gamma distribution posterior of shape two and rate $b>0$, reweighting produces an exponential distribution of rate $b$, so the estimate is $(\log 2)/b$. The weighted first-moment condition ensures finite loss; without it every finite action can have infinite loss, making a median characterization uninformative.