 Exponential posterior for a uniform endpoint

For one observation from a continuous uniform distribution on $[0,\theta ]$ and prior density $\theta e^{-\theta }$, the likelihood cancels the factor $\theta$ and leaves $e^{-\theta }{1}_{\{\theta \ge x\}}$. Normalization gives posterior density $e^{-(\theta -x)}{1}_{\{\theta \ge x\}}$, for $x\ge 0$. Its mean is $x+1$ and variance one. The Bayes estimator under squared error loss is therefore $x+1$, unchanged by multiplying the loss by a positive constant.