Algorithm 1.

Posterior sampling of spatially meshed model (1) and predictions.

	Initialize ${\mathit{\beta }}_j^{(0)}$ and ${\gamma }_j^{(0)}$ for $j=1,...,q$, ${\mathit{w}}_{𝒮}^{(0)}{\mathit{w}}_{\overline{𝒯}}^{(0)}$, and ${\mathit{\theta }}^{(0)}$
	for $t\in \{1,\dots ,T^{\mathrm{*}},T^{\mathrm{*}}+1,\dots ,T^{\mathrm{*}}+T\}$ do	⊳ sequential MCMC loop
1:	for $j=1,...,q$, sample ${\mathit{\beta }}_j^{(t)}\mid {\mathit{y}}_{𝒯},{\mathit{w}}_{𝒯}^{(t-1)},{\gamma }_j^{(t-1)}$
2:	for $j=1,...,q$, sample ${\gamma }_j^{(t)}\mid {\mathit{y}}_{𝒯},{\mathit{w}}_{𝒯}^{(t-1)},{\mathit{\beta }}_j^{(t)}$
3:	sample ${\mathit{\theta }}^{(t)}\mid {\mathit{w}}_{\overline{𝒯}}^{(t-1)},{\mathit{w}}_{𝒮}^{(t-1)}$
	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:	sample ${\mathit{w}}_i^{(t)}\mid {\mathit{w}}_{\mathrm{m}\mathrm{b}(i)}^{(t)},{\mathit{y}}_i,{\mathit{\theta }}^{(t)},{\{{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)}\}}_{j=1}^q$	⊳ reference sampling
	for $\ell \in \overline{𝒯}$ do in parallel
5:	sample $\mathit{w}(\ell {)}^{(t)}\mid {\mathit{w}}_{[\ell ]}^{(t-1)},\mathit{y}(\ell ),{\mathit{\theta }}^{(t)},{\{{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)}\}}_{j=1}^q$	⊳ non-reference sampling
	Assuming convergence has been attained after $T^{\mathrm{*}}$ iterations:
	discard ${\{{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)}\}}_{j=1}^q,{\mathit{w}}_{𝒮}^{(t)},{\mathit{w}}_{\overline{𝒯}}^{(t)},{\mathit{\theta }}^{(t)}$ for $t=1,\dots ,T^{\mathrm{*}}$
	Output: Correlated sample of size $T$ with density
	${\{{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)}\}}_{j=1}^q,{\mathit{w}}_{𝒮}^{(t)},{\mathit{w}}_{\overline{𝒯}}^{(t)},{\mathit{\theta }}^{(t)}\sim {\pi }_{𝒢}({\left\{{\mathit{\beta }}_j,{\gamma }_j\right\}}_{j=1}^q,{\mathit{w}}_{𝒮}^{(t)},{\mathit{w}}_{\overline{𝒯}}^{(t)},\mathit{\theta }\mid {\mathit{y}}_{𝒯})$.
	Predict at ${\ell }^{\mathrm{*}}\in 𝒰$: for $t=1,...,T$ and $j=1,...,q$, sample from $\pi ({\mathit{w}}_{{\ell }^{\mathrm{*}}}^{(t)}\mid {\mathit{w}}_{\left[{\ell }^{\mathrm{*}}\right]}^{(t)},{\mathit{\theta }}^{(t)})$, then from $F_j(w_j{({\ell }^{\mathrm{*}})}^{(t)},{\mathit{\beta }}_j^{(t)},{\gamma }_j^{(t)})$