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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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