Bounded uniform-location posterior ($\theta \mid \mathbf{x}\sim \mathrm{Unif}(\max (A,x_{\max }-d),\min (B,x_{\min }+d))$)

For independent observations uniform on $(\theta -d,\theta +d)$ and a uniform prior on $(A,B)$, the likelihood is constant wherever all data-compatible intervals overlap. The posterior is therefore uniform on the displayed intersection, provided it has positive length. Both squared-error and absolute-error Bayes estimates are its midpoint. An empty or zero-length interval has zero marginal likelihood and cannot be normalized into this ordinary posterior density.