ALGORITHM 1.

MFVB algorithm for lagged kernel machine regression.

Initialize: ${\mu }_{q(1/{\sigma }^2)}>0$, ${\mu }_{q(\beta )}=1$, ${\mu }_{q(1/{\tau }^2)}=1$, ${\mu }_{q(1/{\omega }^2)}=1$.

Cycle:

${{\displaystyle {\sum }_{q(h)}\leftarrow \left\{{\mu }_{q(1/{\sigma }^2)}{W}^{\mathit{\top }}W+{\mu }_{q({\sum }_{{\tau }^2,{\omega }^2}^{-1})}\right\}}}^{-1}$

${\mu }_{q(h)}\leftarrow {\mu }_{q(1/{\sigma }^2)}{\displaystyle {\sum }_{q(h)}{W}^{\mathit{\top }}}(Y-X{\mu }_{q(\beta )})$

${\mu }_{q(\beta )}\leftarrow {(X^{\mathit{\top }}X)}^{-1}{X}^{\mathit{\top }}(Y-W{\mu }_{q(h)})$

${\mu }_{q(1/{\sigma }^2)}\leftarrow \frac{n+2a}{{(Y-X{\mu }_{q(\beta )}-W{\mu }_{q(h)})}^{\mathit{\top }}(Y-X{\mu }_{q(\beta )}-W{\mu }_{q(h)})+2\gamma }$

${\mu }_{q(1/{\tau }_t^2)}\leftarrow {\left\{\frac{{\mu }_{q({\lambda }_1^2)}}{{\mu }_{q({\Vert h_t\Vert }_{G_t}^2}}\right\}}^{1/2}$

${\mu }_{q(1/{\omega }_t^2)}\leftarrow {\left\{\frac{{\mu }_{q({\lambda }_2^2)}}{{\mu }_{q({\displaystyle {\sum }_{n=1}^N{(h_{t+1,n}-h_{t,n})}^2)}}}\right\}}^{1/2}$

${\mu }_{q({\lambda }_1^2)}\leftarrow \frac{\mathit{\top }(n+1)+2r_1}{{\displaystyle {\sum }_{t=1}^{\mathit{\top }}{\mu }_{q({\tau }_t^2)}+2{\delta }_1}}$

${\mu }_{q({\lambda }_2^2)}\leftarrow \frac{\mathit{\top }-1+r_2}{\frac{1}{2}{\displaystyle {\sum }_{t=1}^{\mathit{\top }-1}{\mu }_{q({\omega }_t^2)}+{\delta }_2}}$

until the increase in $\log \{\underset{\_}{p}(\mathbf{y};q)\}$ is negligible.