You are given a prior or prior hyperparameters for a Gaussian process, and observational data, and values \(z\) and \(s\). Find (if relevant) the posterior (predictive) probability distribution of \(\mu, \tau\), and \(X_{+}\). Further, calculate (if relevant) the probabilities \(P(\mu

Unknown \(\mu\), known \(\sigma\).

(a) Prior: \(\kappa_{0}=0, \Sigma_{0}=0 . s_{0}=\) 2.5. Statistics: \(n=5, \Sigma_{x}=23.43 . z=6\).

(b) Prior: \(\kappa_{0}=0, \Sigma_{0}=0 . s_{0}=6\). We have 7 observations with \(\bar{x}=45.4143\). \(z=40\).

(c) Prior: \(\kappa_{0}=0, \Sigma_{0}=0 . s_{0}=15.9\). Data: \(\{319.3,369.4,327.3\} . z=350\).

(d) Prior: \(\kappa_{0}=0, \Sigma_{0}=0 . s_{0}=200\). Statistics: \(n=12, \Sigma_{x}=35723.7\). \(z=3125\).

(e) Prior: \(\phi_{(5,1)} \cdot s_{0}=2\). We have 8 observations with \(\bar{x}=5.45 . z=6\).

(f) Prior: \(\phi_{(510,15)} \cdot s_{0}=30\). Statistics: \(n=7, \Sigma_{x}=3568.2 . z=500\).

(g) Prior: \(\phi_{(0.19,0.03)} \cdot s_{0}=0.03\). Data: \(\{0.174,0.144,0.255,0.237\} . z=0.15\).

Known \(\mu\), unknown \(\sigma\).

(h) Prior: \(v_{0}=-1, S S_{0}=0 . m_{0}=23\). Statistics: \(n=5, \Sigma_{x}=119.03, S S_{x}=\) 3.433 77. \(z=25\) and \(s=1.25\).

(i) Prior: \(v_{0}=-1, S S_{0}=0 . m_{0}=50\). We have 8 observations with average \(\bar{x}=49.8716\) and sample standard deviation \(s_{x}=3.42429 . z=45\) and \(s=10\).
(j) Prior: \(v_{0}=2, S S_{0}=0.16 . m_{0}=1.2\). Data: \(\{1.18,1.14,1.22,1.25,1.15\), \(1.18,1.22,1.15,1.24,1.26\} . z=1\) and \(s=0.1\).\section*{Unknown \(\mu\), unknown \(\sigma\).}
(k) Prior: \(\kappa_{0}=0, \Sigma_{0}=0, v_{0}=-1, S S_{0}=0\). Statistics: \(n=8, \Sigma_{x}=31832\), \(\Sigma_{x^{2}}=126742346 . z=4000\) and \(s=100\).
(I) Prior: \(\kappa_{0}=0, \Sigma_{0}=0, v_{0}=-1, S S_{0}=0\). We have 7 observations with average \(\bar{x}=11.5286\) and sample standard deviation \(s_{x}=24.2342 . z=0\) and \(s=15\).
(m) Prior: \(\kappa_{0}=1, \Sigma_{0}=0.007, v_{0}=1, S S_{0}=1.042\). Data: \(\{0.232,-1.587\), \(-0.986\} . z=0\) and \(s=0.5\).
(n) Prior: \(\kappa_{0}=0, \Sigma_{0}=0, v_{0}=-1, S S_{0}=0\). Statistics: \(n=13, \Sigma_{x}=295.839\), \(S S_{x}=174.964 . z=20\) and \(s=5\).
(o) Prior: \(\kappa_{0}=2, \Sigma_{0}=40, v_{0}=2, S S_{0}=3\). We have 8 observations with average \(\bar{x}=19.6719\) and sample standard deviation \(s_{x}=1.04414 . z=\) 17.9 and \(s=1.179\).
(p) Prior: \(\kappa_{0}=0, \Sigma_{0}=0, v_{0}=-1, S S_{0}=0\). Statistics: \(n=8, \Sigma_{x}=737.1\), \(S S_{x}=6604.9 . z=50\) and \(s=25\).
(q) Prior: \(\kappa_{0}=6, \Sigma_{0}=0.0000198, v_{0}=5, S S_{0}=1.5 \cdot 10^{-7}\). We have 30 observations with average \(\bar{x}=3.30635 \cdot 10^{-6}\) and sample standard deviation \(s_{x}=2.99756 \cdot 10^{-8} . z=-0.00004\) and \(s=0.00004\).