Let $[X_i|\theta] \sim N(0,\theta)$ and suppose $\theta$ has prior distribution:

$$\pi(\theta) = \theta^{-2}e^{-1/\theta}$$

I want to find the Bayes estimator of $\theta$ under square error loss, which I know is given by the posterior mean $E[\theta |X]$. The problem I am having is how to evaluate this expectation. Multipling the above densities we have that the joint density $(X,\theta)$ is given by

$$f(X,\theta) = (2\pi)^{-n/2}\theta^{-n/2-2}\exp \{\frac{-1}{2\theta^2}\sum_{i=1}^n X_i^2 -\frac{1}{\theta} \}$$

But I'm unsure how to proceed further. I know I need to integrate out $\theta$ to obtain the marginal distribution, but I am unsure how to integrate this?

I tried doing integration by parts but just got lost in the process.

  • $\begingroup$ 0. Assume that $\theta$ in the first equation is a variance not a standard deviation. 1. Identify the standard posterior in $\theta$ [or $1/\theta$] from the joint; 2. compute the expectation. $\endgroup$ – Xi'an May 26 at 15:59
  • $\begingroup$ @Xi'an thanks! I just assumed it was SD. Much appreciated $\endgroup$ – Xiaomi May 27 at 0:52

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