Describe the Weak Sufficiency Principle (WSP) and the Strong Sufficiency Principle (SSP). Show that Bayesian inference with a fixed prior distribution respects WSP.

A parameter $\Phi$ has a prior distribution which is normal with mean 0 and precision (inverse variance) $h_{\Phi} .$ Given $\Phi=\phi$, further parameters $\Theta:=\left(\Theta{ }_{i}: i=1, \ldots, I\right)$ have independent normal distributions with mean $\phi$ and precision $h_{\Theta}$. Finally, given both $\Phi=\phi$ and $\boldsymbol{\Theta}=\boldsymbol{\theta}:=\left(\theta_{1}, \ldots, \theta_{I}\right)$, observables $\boldsymbol{X}:=\left(X_{i j}: i=1, \ldots, I ; j=1, \ldots, J\right)$ are independent, $X_{i j}$ being normal with mean $\theta_{i}$, and precision $h_{X}$. The precision parameters $\left(h_{\Phi}, h_{\Theta}, h_{X}\right)$ are all fixed and known. Let $\overline{\boldsymbol{X}}:=\left(\bar{X}_{1}, \ldots, \bar{X}_{I}\right)$, where $\bar{X}_{i}:=\sum_{j=1}^{J} X_{i j} / J$. Show, directly from the definition of sufficiency, that $\overline{\boldsymbol{X}}$ is sufficient for $(\Phi, \Theta)$. [You may assume without proof that, if $Y_{1}, \ldots, Y_{n}$ have independent normal distributions with the same variance, and $\bar{Y}:=n^{-1} \sum_{i=1}^{n} Y_{i}$, then the vector $\left(Y_{1}-\bar{Y}, \ldots, Y_{n}-\bar{Y}\right)$ is independent of $\bar{Y}$.]

For data-values $\boldsymbol{x}:=\left(x_{i j}: i=1, \ldots, I ; j=1, \ldots, J\right)$, determine the joint distribution, $\Pi_{\phi}$ say, of $\Theta$, given $\boldsymbol{X}=\boldsymbol{x}$ and $\Phi=\phi .$ What is the distribution of $\Phi$, given $\boldsymbol{\Theta}=\boldsymbol{\theta}$ and $\boldsymbol{X}=\boldsymbol{x} ?$

Using these results, describe clearly how Gibbs sampling combined with RaoBlackwellisation could be applied to estimate the posterior joint distribution of $\Theta$, given $\boldsymbol{X}=\boldsymbol{x}$.