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Let $X_i$ for $i=1,\dots,n$ be i.i.d. normal $N(\mu,\sigma^2)$ where $\sigma$ is known. Use the Gaussian prior on $\mu$, $g(\mu)=N(\theta,\tau^2)$ and find the Bayes estimator of $\mu$.

For $X_{1},X_{2},...,X_{n}$ iid $N(\mu,\sigma^2)$, and a priori distribution $\mu\sim N(\theta,\tau^2)$, I think I should obtain a posteriori distribution $N(\theta_{\ast},\tau^2_{\ast})$, where:

$$\theta_{\ast}=\frac{\frac{n}{\sigma^2}\bar{x}+\frac{\theta}{\tau^2}}{\frac{n}{\sigma^2}+\frac{1}{\tau^2}}\quad\text{and}\quad\tau^{2}_{\ast}=\left(\frac{n}{\sigma^2}+\frac{1}{\tau^2}\right)^{-1}$$

from reading here http://en.wikipedia.org/wiki/Conjugate_prior. However, as for the Bayesian estimator - well, I believe that that would depend on my risk function as the Bayes estimaor is given by the mean of prosterior; with a MSE function, I think I should obtain $\theta^{B}_{\Pi}=\mu_{\ast}$.

Am I correct? If so is there some rule of thumb where I should just assume a MSE function? We haven't talked much about any Bayesian analysis in my course, so I am quite confused.

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You're correct.

Regarding a rule of thumb on when to use Bayesian MSE, it really depends on your posterior distribution.

The posterior you obtained in the question is normal and hence its unimodal and further more mean = median = mode.

You would like to use some other bayes estimator (derived for a different loss function), for example, if you had a multimodal posterior.

I would really emphasize on the depends on the posterior part. I would really emphasize

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