Exercise 6.2 - James Stein estimator for Gaussian means

The prior for ${𝜃}_i$ is:

and the likelihood is given by:

For question (a), we begin by integrating out ${𝜃}_i$ and establishing the dependency of $𝒟={\left\{y_i\right\}}_{i=1}^6$ on $m_0$ and ${\tau }_0^2$:

where we have canceled terms independent from $y$ and completing the square in the final step of deduction. Given ${\sigma }^2=500$, we have:

For question (b), the posterior distribution of ${𝜃}_i$ given $y_i$ and other hyperparameters is:

For question (c), the interval is $({\mu }^{\prime }-1.96{\sigma }^{\prime },{\mu }^{\prime }+1.96{\sigma }^{\prime })$, where ${\mu }^{\prime }$ and ${\sigma }^{\prime }$ are from question (b).

For question (d), a smaller ${\sigma }^2$ would reduce the ML-II into the ordinary posterior analysis. The parameter ${\sigma }^2$ can be understood as the noise on the observations. The less noise we assumed, the more precise the observations are, and the intermedia $𝜃$ becomes less necessary.