Algorithm 1.

Iterative scheme for obtaining the parameters in the optimal densities in the longitudinal functional regression model (2.8).

Initialize: $B_{q({\sigma }_{\mathit{g}}^2)}\dots B_{q({\sigma }_{\mathit{Y}}^2)}>0$, , μq(C) = 0, μq(g) = 0, μq(β) = 0, Σq(g) = I, Λq = I.

Cycle:

$$\begin{array}{l}{\mathbf{\sum }}_{q(\mathit{\beta })}\leftarrow {\left\{{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}{\mathit{z}}^T\mathit{z}+\frac{1}{{\sigma }_{\mathit{\beta }}^2}\mathit{I}\right\}}^{-1}\\ {\mathit{\mu }}_{q(\mathit{\beta })}\leftarrow {\mathbf{\sum }}_{q(\mathit{\beta })}{\left\{{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}\left({\mathit{Y}}^T-{\mathit{\mu }}_{q(\mathit{b})}^T{\mathit{Z}}^T-{\mathit{\mu }}_{q(\mathit{g})}^T\mathit{M}{\mathit{\mu }}_{q(\mathit{C})}^T\right)\mathit{z}\right\}}^T\\ {\mathbf{\sum }}_{q(\mathit{b})}\leftarrow {\left\{{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}{\mathit{Z}}^T\mathit{Z}+{\mu }_{q(1/{\sigma }_{\mathit{b}}^2)}\mathit{I}\right\}}^{-1}\\ {\mathit{\mu }}_{q(\mathit{b})}\leftarrow {\mathbf{\sum }}_{q(\mathit{b})}{\left\{{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}\left({\mathit{Y}}^T-{\mathit{\mu }}_{q(\mathit{\beta })}^T{\mathit{z}}^T-{\mathit{\mu }}_{q(\mathit{g})}^T\mathit{M}{\mathit{\mu }}_{q(\mathit{C})}^T\right)\mathit{Z}\right\}}^T\\ {\mathbf{\sum }}_{q(\mathit{C})}\leftarrow {\left\{{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}\mathit{M}({\mathit{\mu }}_{q(\mathit{g})}{\mathit{\mu }}_{q(\mathit{g})}^T+{\mathbf{\sum }}_{q(\mathit{g})}){\mathit{M}}^T+{\mu }_{q(1/{\sigma }_{\mathit{X}}^2)}{\mathit{\psi }}^T\mathit{\psi }+{\mathrm{\Lambda }}_q^{-1}\right\}}^{-1}\\ {\mathit{\mu }}_{q(\mathit{C})}^T\leftarrow {\mathbf{\sum }}_{q(\mathit{C})}{({\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}Y{\mathit{\mu }}_{q(\mathit{g})}^T{\mathit{M}}^T-{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}{\mu }_{q(\mathit{\beta })}\mathit{z}{\mathit{\mu }}_{q(\mathit{g})}^T{\mathit{M}}^T{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}\mathit{Z}{\mathit{\mu }}_{q(\mathit{b})}{\mathit{\mu }}_{q(\mathit{g})}^T{\mathit{M}}^T+{\mu }_{q(1/{\sigma }_{\mathit{X}}^2)}\mathit{W}\mathit{\psi })}^T\\ {\mathbf{\sum }}_{q(\mathit{g})}\leftarrow {\left\{{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}{\mathit{M}}^T({\mathit{\mu }}_{q(\mathit{C})}^T{\mathit{\mu }}_{q(\mathit{C})}+n{\mathbf{\sum }}_{q(\mathit{C})})\mathit{M}+{\mu }_{q(1/{\sigma }_{\mathit{g}}^2)}{\mathit{D}}^{-1}\right\}}^{-1}\\ {\mathit{\mu }}_{q(\mathit{g})}\leftarrow {\mathbf{\sum }}_{q(\mathit{g})}{\left\{{\mu }_{q(1/{\sigma }_{\mathit{Y}}^2)}\left({\mathit{Y}}^T-{\mathit{\mu }}_{q(\mathit{\beta })}^T{\mathit{z}}^T-{\mathit{\mu }}_{q(\mathit{b})}^T{\mathit{Z}}^T\right)({\mathit{\mu }}_{q(\mathit{C})}\mathit{M})\right\}}^T\\ B_{q({\lambda }_k)}\leftarrow B_{\lambda }+\frac{1}{2}\left({({\mathit{\mu }}_{q(\mathit{C})}^k)}^T({\mathit{\mu }}_{q(\mathit{C})}^k)+n{({\mathbf{\sum }}_{q(\mathit{C})})}_{kk}\right),1\le j\le K_x\\ B_{q({\sigma }_{\mathit{X}}^2)}\leftarrow B_{\mathit{X}}+\frac{1}{2}\left\{{\sum }_{i=1}^I{\sum }_{j=1}^J{\left|\right|{({\mathit{W}}_{ij}-{\mathit{\mu }}_{q(\mathit{c}),ij}{\mathit{\psi }}^T)}^T\left|\right|}^2+(nJ)\text{tr}({\mathit{\psi }}^T\mathit{\psi }{\mathbf{\sum }}_{q(\mathit{C})})\right\}\\ B_{q({\sigma }_{\mathit{b}}^2)}\leftarrow B_{\mathit{b}}+\frac{1}{2}\left({\mathit{\mu }}_{q(\mathit{b})}^T{\mathit{\mu }}_{q(\mathit{b})}+\text{tr}({\mathbf{\sum }}_{q(\mathit{b})})\right)\\ B_{q({\sigma }_{\mathit{g}}^2)}\leftarrow B_{\mathit{g}}+\frac{1}{2}\left({\mathit{\mu }}_{q(\mathit{g})}^T{\mathit{D}}^{-1}{\mathit{\mu }}_{q(\mathit{g})}+\text{tr}({\mathit{D}}^{-1}{\mathbf{\sum }}_{q(\mathit{g})})\right)\\ B_{q({\sigma }_{\mathit{Y}}^2)}\leftarrow B_{\mathit{Y}}+\frac{1}{2}[{\left|\right|\mathit{Y}-\mathit{z}{\mathit{\mu }}_{q(\mathit{\beta })}-\mathit{Zb}-{\mathit{\mu }}_{q(\mathit{C})}\mathit{M}{\mathit{\mu }}_{q(\mathit{g})}\left|\right|}^2+\text{tr}({\mathit{z}}^T\mathit{z}{\mathbf{\sum }}_{q(\mathit{\beta })})\\ -{({\mathit{\mu }}_{q(\mathit{g})}\mathit{M}{\mathit{\mu }}_{q(\mathit{C})})}^T({\mathit{\mu }}_{q(\mathit{g})}\mathit{M}{\mathit{\mu }}_{q(\mathit{C})})\\ +{\mathit{\mu }}_{q(\mathit{g})}^T{\mathit{M}}^T({\mathit{\mu }}_{q(\mathit{C})}^T{\mathit{\mu }}_{q(\mathit{C})}+(nJ){\mathbf{\sum }}_{q(\mathit{C})})\mathit{M}{\mathit{\mu }}_{q(\mathit{g})}\\ +\text{tr}\left\{{\mathit{M}}^T\left({\mathit{\mu }}_{q(\mathit{C})}^T{\mathit{\mu }}_{q(\mathit{C})}+(nJ){\mathbf{\sum }}_{q(\mathit{C})}\right)\mathit{M}{\mathbf{\sum }}_{q(\mathit{g})}\right\}\\ +\text{tr}\left\{{\mathit{Z}}^T\mathit{Z}{\mathbf{\sum }}_{q(\mathit{b})}\right\}]\end{array}$$

until the increase in p(Y, W; q) is negligible.