Consider a Bayesian model where the prior distribution is $\lambda(\theta) = \frac{2}{(1+\theta)^3}$, $\theta > 0$ and $X\sim U(0,\theta).$ Find the Bayes estimate $\delta(x)$ if the loss function is $L(\theta, d) = |d-\theta|.$
I know that $\theta\mid X\sim f(x, \theta) = \frac{2}{\theta(1+\theta)^3}$ and the bayes estimate under absolute loss function is the median of the posterior distribution. However, I got stuck on finding the median. Can anyone help me out? Thank you.
The model is uniform in $[0;1]$ so the posterior is
$$f(\theta\mid\mathbf{x})=\frac{2}{(1+\theta)^3}$$
To calculate the median is enough to calculate
$$\int_0^\theta \frac{2}{(1+u)^3}\,du=\frac{1}{2}$$
That easily leads to $\hat{\theta}=\sqrt{2}-1$