Algorithm 2: HPO EM algorithm for the diabetes dataset
1. Generate randomly hyperparameters value $\eta$, $\lambda$, $\mu$.
2. Choose a suitable linear basis function $\psi (x_n)$ to get an $8\times 442$ matrix $\Psi$ by using (12).
3. E step. Compute the mean $m$ and covariance $C$ using the current hyperparameter values.
$m={(\Lambda +\lambda {\Psi }^T\Psi )}^{-1}(\Lambda \mu +\lambda {\Psi }^{T}T),$	(54)
$C={(\Lambda +\lambda {\Psi }^T\Psi )}^{-1}.$	(55)
4. M step. Estimate again the hyperparameters by employing the mean $m$ and covariance $C$ obtained by step 3 and the following update equations
${\eta }_i^{new}={\displaystyle \frac{1}{m_i^2+C_{ii}}},$	(56)
${\lambda }^{new}={\displaystyle \frac{N}{{‖T-\Psi m‖}^2+Tr({\Psi }^T\Psi C)}},$	(57)
${\mu }^{new}=m.$	(58)
5. Compute the likelihood function or log likelihood function given by the following result:
$p(T|X,\eta ,\lambda ,\mu )=\sqrt{{\left({\displaystyle \frac{\lambda }{2\pi }}\right)}^N}{\left|\Lambda \right|}^{{\displaystyle \frac{1}{2}}}{\left|C\right|}^{{\displaystyle \frac{1}{2}}}\exp (-G(m))$
or
$\ln p(T|X,\eta ,\lambda ,\mu )={\displaystyle \frac{N}{2}}\ln \lambda -G(m)+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^M\ln }{\eta }_i+{\displaystyle \frac{1}{2}}\ln \left|C\right|-{\displaystyle \frac{N}{2}}\ln (2\pi )$
and then determine the convergence of the hyperparameters or the likelihood. If convergence criterion is not satisfied, go back to step 3.