Algorithm 3.

Posterior sampling and prediction of LMC model (1) with MGP priors.

	Initialize ${\mathit{\beta }}_j^{(0)},{\mathbf{\Lambda }}^{(0)}$ and ${\gamma }_j^{(0)}$ for $j=1,\dots ,q,{\mathit{v}}_{𝒮}^{(0)}$, and ${\mathbf{\Phi }}^{(0)}$
	for $t\in \{1,\dots ,T^{\mathrm{*}},T^{\mathrm{*}}+1,\dots ,T^{\mathrm{*}}+T\}$ do	⊳ sequential MCMC loop
	for $j=1,...,q$, do in parallel
1:	use SiMPA to update ${\mathit{\beta }}_j^{(t)},{\mathit{\lambda }}_{[j;:]}^{(t)}\mid {\mathit{y}}_{𝒯},{\mathit{v}}_{𝒮}^{(t-1)},{\gamma }_j^{(t-1)}$	⊳ $▹O(nq(p+k{)}^2)$
	for $j=1,...,q$, do in parallel
2:	use Metropolis-Hastings to update ${\gamma }_j^{(t)}\mid {\mathit{y}}_{𝒯},{\mathit{v}}_{𝒮}^{(t-1)},{\mathit{\beta }}_j^{(t)},{\mathit{\lambda }}_{[j,:]}^{(t)}$	⊳ $O(nq)$
3:	use Metropolis-Hastings to update ${\mathbf{\Phi }}^{(t)}\mid {\mathit{v}}_{𝒮}^{(t-1)}$	⊳ $O\left(nk{d}^3{m}^2\right)$
	for $c\in \mathrm{C}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{r}\mathrm{s}(𝒢)$ do	⊳ sequential
	for $i\in \left\{i:\mathrm{C}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{r}\left(a_i\right)=c\right\}$ do in parallel
4:	use SiMPA to update ${\mathit{v}}_i^{(t)}\mid {\mathit{v}}_{\mathrm{m}\mathrm{b}(i)}^{(t)},{\mathit{y}}_i,{\mathbf{\Lambda }}^{(t)},{\mathbf{\Phi }}^{(t)},{\{{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)}\}}_{j=1}^q$	⊳ $O\left(nm{k}^2\right)$
	Assuming convergence has been attained after $T^{\mathrm{*}}$ iterations:
	discard ${\{{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)}\}}_{j=1}^q,{\mathit{v}}_{𝒮}^{(t)},{\mathbf{\Lambda }}^{(t)},{\mathbf{\Phi }}^{(t)}$ for $t=1,\dots ,T^{\mathrm{*}}$
	Output: Correlated sample of size $T$ with density
	${\{{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)}\}}_{j=1}^q,{\mathit{v}}_{𝒮}^{(t)},{\mathbf{\Lambda }}^{(t)},{\mathbf{\Phi }}^{(t)}\sim {\pi }_{𝒢}({\left\{{\mathit{\beta }}_j,{\gamma }_j\right\}}_{j=1}^q,{\mathit{v}}_{𝒮}^{(t)},\mathbf{\Lambda },\mathbf{\Phi },\mid {\mathit{y}}_{𝒯})$.
	Predict at ${\ell }^{\mathrm{*}}\in 𝒰:$ for $t=1,...,T$ and $j=1,...,q,$ sample from $\pi ({\mathit{v}}_{{\ell }^{\mathrm{*}}}^{(t)}\mid {\mathit{v}}_{[{\ell }^{\mathrm{*}}]}^{(t)},{\mathbf{\Phi }}^{(t)})$, then from $F_j(w_j{({\ell }^{\mathrm{*}})}^{(t)},{\mathit{\beta }}_j^{(t)},{\mathit{\lambda }}_{[j,:]}^{(t)},{\gamma }_j^{(t)})$