 Solution

For an observed $x\ge 0$, the continuous uniform distribution likelihood is $p(x\mid \theta )={\theta }^{-1}{1}_{\{\theta \ge x\}}$, with $\theta >0$. Multiplying by the stated prior cancels the factor $\theta$, giving $e^{-\theta }{1}_{\{\theta \ge x\}}$. Its integral over the permitted parameter values is $e^{-x}$. Hence the posterior distribution isThis is the exponential posterior for a uniform endpoint. Its posterior mean is $x+1$ and its variance is one. The posterior expected squared-error loss decomposes asSince $c>0$, its unique minimum occurs at the Bayes estimate $\boxed{{\hat{\theta }}_B=x+1}$. The positive loss multiplier changes the risk, not the optimizing action.