This is for a personal review: $X_i\sim U(0,\theta), \theta\sim$Gamma($n+1, b$), $\theta>0$. Want to find Bayes estimate assuming the loss function $L(\theta, d(x)) = (\theta-d(x))^2.$ The Gamma has a shape parameter $n+1$ and scale paramter $b$
As we know the posterior distribution is proportional to the prior and the likelihood function: $$\tau(\theta|x) \propto \tau(\theta) \cdot \tau(x|\theta) = \theta e^{-\theta/b} \cdot \mathbb{I} (\max_{1\le i\le n} x_i \le \theta).$$ I am not sure if I can call this exponential($\theta$) so that the Bayes estimate becomes the mean of $\tau(\theta|x)$ which is $1/\theta.$ However, I think I'm missing something about $\mathbb{I} (\max_{1\le i \le n} x_i).$ Help is needed. Thanks in advance!