Algorithm 5 PCGS in the case of a scale mixture of Gaussian noise.

Initialize: ${\mathbf{x}}^{(0)}\in {\mathbb{R}}^Q$, ${\mathbf{v}}^{(0)}\in {\mathbb{R}}^Q$, ${\mathbf{\sigma }}^{(0)}\in {\mathbb{R}}_{+}^N$, ${\mathsf{\Theta }}^{(0)}\in {\mathbb{R}}^P$ 1:for $t=0,1,\dots$ do 2: For all $i\in \{1,\dots ,N\}$, generate ${\sigma }_i^{(t+1)}\sim {\mathcal{P}}_{{\sigma }_i|{\mathbf{x}}^{(t)},{\mathsf{\Theta }}^{(t)},\mathbf{z}}$ 3: Generate ${\mathsf{\Theta }}^{(t+1)}\sim {\mathcal{P}}_{\mathsf{\Theta }|{\mathbf{x}}^{(t)},{\mathbf{\sigma }}^{(t+1)},\mathbf{z}}$ 4: Set $\mu ({\mathbf{\sigma }}^{(t)})$ and $\mathsf{\Gamma }({\mathbf{\sigma }}^{(t)})$ 5: Generate ${\mathbf{v}}^{(t+1)}\sim {\mathcal{P}}_{\mathbf{v}|{\mathbf{x}}^{(t)},{\mathbf{\sigma }}^{(t+1)},{\mathsf{\Theta }}^{(t+1)},\mathbf{z}}$ 6: Generate ${\mathbf{x}}^{(t+1)}\sim {\mathcal{P}}_{\mathbf{x}\mid {\mathbf{v}}^{(t+1)},{\mathbf{\sigma }}^{(t+1)},{\mathsf{\Theta }}^{(t+1)},\mathbf{z}}$ 7:end for