Table 1.

Distributions for the nodes of the graphical model obtained using Eq. (47). Derivations are shown in Appendix D, where matrices Q, P₁, P₂, and W(u) and the cov(·) operator are defined. The matrix R_{(k, q)} is a kq × kq permutation matrix with the property R_{(k, q)}vec(Z^T)=vec(Z) (the matrix R_{(t1, q)} is defined analogously).

Functional form	Parameters
q(S) = 𝒩(vec(S)|vec(〈S〉),It1⊗ΣS)	〈S〉 = ΣS(〈α1〉LTM + 〈∊1〉Diag(〈w〉)C〈X〉)
	ΣS = (〈α1〉LTL + 〈∊1〉In)−1
q(H) = 𝒩(vec(H)|vec(〈H〉),In⊗ΣH)	〈H〉 = ΣH(〈α2〉BTY + 〈∊2〉〈Z〉TCTDiag (〈w〉))
	ΣH = (〈α2〉BTB + 〈∊2〉Ik)−1
q(X)= 𝒩(vec(X)|vec(〈X〉), Σx)	vec(〈X〉) = 〈∊1〉 Σx(It1⊗CTDiag(〈w〉))vec(〈S〉)
	${\Sigma }_{\mathbf{X}}={\left(\langle {∊}_1\rangle ({\mathbf{I}}_{t_1}\otimes \mathbf{Q})+{\mathbf{R}}_{(t_1,q)}^T\left(\text{Diag}\left(\langle {\mathrm{\beta }}_1\rangle \right)\otimes {\mathbf{T}}_1^T{\mathbf{T}}_1\right){\mathbf{R}}_{(t_1,q)}\right)}^{-1}$
q(Z) = 𝒩(vec(Z) | vec(〈Z〉), ΣZ)	vec(〈Z〉) = 〈∊2〉 ΣZ(Ik⊗CTDiag(〈w〉))vec(〈H〉T)
	${\Sigma }_{\mathbf{Z}}={\left(\langle {∊}_2\rangle ({\mathbf{I}}_k\otimes \mathbf{Q})+{\mathbf{R}}_{(k,q)}^T\left(\text{Diag}\left(\langle {\mathrm{\beta }}_2\rangle \right)\otimes {\mathbf{T}}_2^T{\mathbf{T}}_2\right){\mathbf{R}}_{(k,q)}\right)}^{-1}$
q(w) = 𝒩(w | 〈w〉, Σw)	〈w〉 = Σwdiag(〈∊1〉 〈S〉 〈X〉TCT + 〈∊2〉〈H〉T 〈Z〉TCT)
	Σw = (〈∊1〉 P1 + 〈∊2〉 P2 + 〈γ〉W(u))−1
q(α1)= Γ(α1|aα1, bα1)	$a_{{\alpha }_1}=\frac{{\mathit{mt}}_1}{2}+a_{{\alpha }_1}^0$
	$b_{{\alpha }_1}=\frac{1}{2}\mathrm{tr}\left({(\mathbf{M}-\mathbf{L}\langle \mathbf{S}\rangle )}^T(\mathbf{M}-\mathbf{L}\langle \mathbf{S}\rangle )\right)+\frac{t_1}{2}\mathrm{tr}\left({\Sigma }_{\mathbf{S}}{\mathbf{L}}^T\mathbf{L}\right)+b_{{\alpha }_1}^0$
q(α2) = Γ(α2 | aα2, bα2)	$a_{{\alpha }_2}=\frac{{\mathit{nt}}_2}{2}+a_{{\alpha }_2}^0$
	$b_{{\alpha }_2}=\frac{1}{2}\mathrm{tr}\left({(\mathbf{Y}-\mathbf{B}\langle \mathbf{H}\rangle )}^T(\mathbf{Y}-\mathbf{B}\langle \mathbf{H}\rangle )\right)+\frac{n}{2}\mathrm{tr}\left({\Sigma }_{\mathbf{H}}{\mathbf{B}}^T\mathbf{B}\right)+b_{{\alpha }_2}^0$
q(∊1) = Γ(∊1|a∊1, b∊1)	$a_{{∊}_1}=\frac{t_{1}n}{2}$
	$b_{{∊}_1}=\frac{1}{2}\left[\mathrm{tr}\left({\langle \mathbf{S}\rangle }^T\langle \mathbf{S}\rangle -2{\langle \mathbf{S}\rangle }^T\text{Diag}\left(\langle \mathbf{w}\rangle \right)\mathbf{C}\langle \mathbf{X}\rangle +{\langle \mathbf{X}\rangle }^T\mathbf{Q}\langle \mathbf{X}\rangle \right)+t_1\mathrm{tr}\left({\Sigma }_{\mathbf{S}}\right)+\mathrm{tr}\left({\Sigma }_{\mathbf{X}}({\mathbf{I}}_{t_1}\otimes \mathbf{Q})\right)\right]$
q(∊2) = Γ(∊2|a∊1, b∊1)	$a_{{∊}_2}=\frac{\mathit{kn}}{2}$
	$b_{{∊}_2}=\frac{1}{2}\left[\mathrm{tr}\left(\langle \mathbf{H}\rangle {\langle \mathbf{H}\rangle }^T-2\langle \mathbf{H}\rangle \text{Diag}\left(\langle \mathbf{w}\rangle \right)\mathbf{C}\langle \mathbf{Z}\rangle +{\langle \mathbf{Z}\rangle }^T\mathbf{Q}\langle \mathbf{Z}\rangle \right)+n\mathrm{tr}\left({\Sigma }_{\mathbf{H}}\right)+\mathrm{tr}\left({\Sigma }_{\mathbf{Z}}({\mathbf{I}}_k\otimes \mathbf{Q})\right)\right]$
q((β1)i) = Γ((β1)i|(aβ1)i, (bβ1)i)	$\left({\mathbf{a}}_{{\mathrm{\beta }}_1}\right)=\frac{t_1}{2}+a_{{\mathrm{\beta }}_1}^0$
	${\left({\mathbf{b}}_{{\mathrm{\beta }}_1}\right)}_i=\frac{1}{2}\langle {\mathbf{X}}_i.\rangle {\mathbf{T}}_1^T{\mathbf{T}}_1{\langle {\mathbf{X}}_i.\rangle }^T+\frac{1}{2}\mathrm{tr}\left({\mathbf{T}}_1^T{\mathbf{T}}_1\mathrm{cov}\left({({\mathbf{X}}_i.)}^T\right)\right)+\langle {\mathrm{\delta }}_1\rangle$
q((β2)i) = Γ((β2)i|(aβ2)i, (bβ2)i)	${\left({\mathbf{a}}_{{\mathrm{\beta }}_2}\right)}_i=\frac{k}{2}+a_{{\mathrm{\beta }}_2}^0$
	${\left({\mathbf{b}}_{{\mathrm{\beta }}_2}\right)}_i=\frac{1}{2}\langle {\mathbf{Z}}_i.\rangle {\mathbf{T}}_2^T{\mathbf{T}}_2{\langle {\mathbf{Z}}_i.\rangle }^T+\frac{1}{2}\mathrm{tr}\left({\mathbf{T}}_2^T{\mathbf{T}}_2\mathrm{cov}\left({({\mathbf{Z}}_i.)}^T\right)\right)+\langle {\mathrm{\delta }}_2\rangle$
q(δ1) = Γ(δ1|aδ1, bδ1)	aδ1 = aβ10q
	$b_{{\mathrm{\delta }}_1}={\Sigma }_{i=1}^q\langle {\left({\mathrm{\beta }}_1\right)}_i\rangle$
q(δ2)= Γ(δ2|aδ2, bδ2)	aδ2 = aβ20q
	$b_{{\mathrm{\delta }}_2}={\Sigma }_{i=1}^q\langle {\left({\mathrm{\beta }}_2\right)}_i\rangle$
q(γ) = Γ(γ|aγ, bγ)	aγ = φn
	$b_{\mathrm{\gamma }}={\Sigma }_{i=1}^n\sqrt{u_i}$