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

Under loss $w(\theta )|\theta -a|$, assume $0<\mathbb{E}[w(\theta )\mid x]<\infty$. The posterior expected loss is a positive constant times ordinary absolute-error loss under the probability density proportional to $w(\theta )\pi (\theta \mid x)$. Its minimizers are the medians of that weighted posterior: slopes of the loss change sign where the weighted probability below $a$ crosses one half. In particular, relative absolute loss has $w(\theta )=1/\theta$ for positive parameters. For a Gamma distribution posterior of shape two and rate $b$, the weighted posterior is an exponential distribution of rate $b$, so the estimate is $(\log 2)/b$.