Algorithm 3.

GAR - Full Bayes

Input: ${\mathbb{X}}_n$and W; parameters r,r’, s, s’, K; sets${\left\{A_k\right\}}_{k=1}^K$; start values of Ω,α, and β; number of iterations B.

For b = 1,..., B:

1. For i = 1, ...,d:

${\alpha }_i^2|\text{rest}~\Gamma \left((r+d+1)/2,s+C_i\right)$,

where $C_i={\omega }_{ii}+{\sum }_{j\ne i}{\alpha }_j^2{\sum }_{k=1}^K{\beta }_k^2{I}_{A_k}\left(W_{ij}\right){\omega }_{ij}^2{\tau }_{ij}^{-1}$.

2. For k = 1,..., K:

${\beta }_k^2|\text{rest}~\Gamma \left(\left(r^{\prime }+D_k\right)/2,s^{\prime }+E_k\right)$,

where $D_k={\sum }_{i<j}{I}_{A_k}\left(W_{ij}\right)$, and

$E_k={\sum }_{i<j}{I}_{A_k}\left(W_{ij}\right){\omega }_{ij}^2{\alpha }_i^2{\alpha }_j^2{\tau }_{ij}^{-1}$.

3. Do steps 1–2 of Algorithm 2 with current Λ.

4. Set Ω(b) = Ω and Θ(b) = {α, β}.

Output: Sequences {Ω(1),Θ(1)},…, {Ω(B),Θ(B)}.