Table A.1.

Update rules for computing the posterior q over the unobserved variables.

${\mathrm{\omega ̃}}_{\mathit{\text{jk}}}^r=\text{exp}(E[\text{log }\alpha ]+E[\text{log }{\upsilon }_k]+{\displaystyle {\sum }_{k\prime <k}}E[\text{log}(1-{\upsilon }_{k\prime })])$

$E[r_{\mathit{\text{jk}}}]={\mathrm{\omega ̃}}_{\mathit{\text{jk}}}^r{E}_z[\mathrm{\Psi }({\mathrm{\omega ̃}}_{\mathit{\text{jk}}}^r+n_{\mathit{\text{jk}}})-\mathrm{\Psi }({\mathrm{\omega ̃}}_{\mathit{\text{jk}}}^r)]$

${\upsilon }_k~\text{Beta}({\mathrm{\omega ̃}}_k^{\upsilon ,1},{\mathrm{\omega ̃}}_k^{\upsilon ,2})$

${\mathrm{\omega ̃}}_k^{\upsilon ,1}=1+{\displaystyle {\sum }_j}E[r_{\mathit{\text{jk}}}]$

${\mathrm{\omega ̃}}_k^{\upsilon ,2}=E[\gamma ]+{\displaystyle {\sum }_{j,k\prime >k}}E[r_{\mathit{\text{jk}}\prime }]$

${\varphi }_{k,s}~\text{Beta}({\mathrm{\omega ̃}}_{k,s}^{\varphi ,1},{\mathrm{\omega ̃}}_{k,s}^{\varphi ,2})$

${\mathrm{\omega ̃}}_{k,s}^{\varphi ,1}={\omega }^{\varphi ,1}+{\displaystyle {\sum }_{j,i}}q(z_{\mathit{\text{ji}}}=k)q(x_{\mathit{\text{jis}}}=1)$

${\mathrm{\omega ̃}}_{k,s}^{\varphi ,2}={\omega }^{\varphi ,2}+{\displaystyle {\sum }_{j,i}}q(z_{\mathit{\text{ji}}}=k)q(x_{\mathit{\text{jis}}}=0)$

$a_{\mathit{\text{ji}}}~\text{Normal}({\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{a,1}{({\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{a,2})}^{-1},{({\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{a,2})}^{-1})$

${\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{a,2}={({\sigma }_j^a)}^{-1}+E[{\lambda }_{\mathit{\text{ji}}}]{\displaystyle {\sum }_{s,s\prime }}E[x_{\mathit{\text{jis}}}{x}_{\mathit{\text{jis}}\prime }]\text{Tr}(E[{\mathit{h}}_j{\mathit{h}}_j^t]{\mathrm{\Xi }}_{\mathit{\text{js}}}^t{\mathrm{\Xi }}_{\mathit{\text{js}}\prime })$

${\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{a,1}={\mu }_j^a{\left({\sigma }_j^a\right)}^{-1}+E[{\lambda }_{\mathit{\text{ji}}}]{\displaystyle {\sum }_s}E[x_{\mathit{\text{jis}}}]E{[{\mathit{h}}_j]}^t{\mathrm{\Xi }}_{\mathit{\text{js}}}^t({\mathit{y}}_{\mathit{\text{ji}}}-{\displaystyle {\sum }_d}E[e_{\mathit{\text{jid}}}]{\mathit{f}}_{\mathit{\text{jd}}})$

${\lambda }_{\mathit{\text{ji}}}~\text{Gamma}({\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{\lambda ,1},{\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{\lambda ,2})$

${\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{\lambda ,1}={\kappa }_j+\frac{T_j}{2}$

${\mathrm{\omega ̃}}_{\mathit{\text{ji}}}^{\lambda ,2}={\theta }_j+{\Vert {\mathit{y}}_{\mathit{\text{ji}}}\Vert }^2+{\displaystyle {\sum }_d}(E[e_{\mathit{\text{jid}}}^2]{\Vert {\mathit{f}}_{\mathit{\text{jd}}}\Vert }^2+{\displaystyle {\sum }_{d\prime \ne d}}E[e_{\mathit{\text{jid}}}]E[e_{\mathit{\text{jid}}\prime }]{\mathit{f}}_{\mathit{\text{jd}}}^t{\mathit{f}}_{\mathit{\text{jd}}\prime })+E[a_{\mathit{\text{ji}}}^2]{\displaystyle {\sum }_{s,s\prime }}E[x_{\mathit{\text{jis}}}{x}_{\mathit{\text{jis}}\prime }]\text{Tr}(E[{\mathit{h}}_j{\mathit{h}}_j^t]{\mathrm{\Xi }}_{\mathit{\text{js}}}^t{\mathrm{\Xi }}_{\mathit{\text{js}}\prime })+E[a_{\mathit{\text{ji}}}]{\displaystyle {\sum }_{s,d}}E[e_{\mathit{\text{jid}}}]E[x_{\mathit{\text{jis}}}]{\mathit{f}}_{\mathit{\text{jd}}}^t{\mathrm{\Xi }}_{\mathit{\text{js}}}E[{\mathit{h}}_j]-{\mathit{y}}_{\mathit{\text{ji}}}^t({\displaystyle {\sum }_d}E[e_{\mathit{\text{jid}}}]{\mathit{f}}_{\mathit{\text{jd}}}+E[a_{\mathit{\text{ji}}}]{\displaystyle {\sum }_s}E[x_{\mathit{\text{jis}}}]{\mathrm{\Xi }}_{\mathit{\text{js}}}E[{\mathit{h}}_j])$

$e_{\mathit{\text{jid}}}~\text{Normal}({\mathrm{\omega ̃}}_{\mathit{\text{jid}}}^{e,1}{({\mathrm{\omega ̃}}_{\mathit{\text{jid}}}^{e,2})}^{-1},{({\mathrm{\omega ̃}}_{\mathit{\text{jid}}}^{e,2})}^{-1})$

${\mathrm{\omega ̃}}_{\mathit{\text{jid}}}^{e,2}={({\sigma }_{\mathit{\text{jd}}}^e)}^{-1}+E[{\lambda }_{\mathit{\text{ji}}}]{\Vert {\mathit{f}}_{\mathit{\text{jd}}}\Vert }^2$

${\mathrm{\omega ̃}}_{\mathit{\text{jid}}}^{e,1}={\mu }_{\mathit{\text{jd}}}^e{({\sigma }_{\mathit{\text{jd}}}^e)}^{-1}+E[{\lambda }_{\mathit{\text{ji}}}]{\mathit{f}}_{\mathit{\text{jd}}}^t({\mathit{y}}_{\mathit{\text{ji}}}-{\displaystyle {\sum }_{d\prime \ne d}}E[e_{\mathit{\text{jid}}\prime }]{\mathit{f}}_{\mathit{\text{jd}}\prime }-E[a_{\mathit{\text{ji}}}]{\displaystyle {\sum }_s}E[x_{\mathit{\text{jis}}}]{\mathrm{\Xi }}_{\mathit{\text{js}}}E[{\mathit{h}}_j])$

$h_j~\text{Normal}({\mathrm{\Xi }}_j^{-1}{\mathit{\omega ̃}}_j^h,{\mathrm{\Xi }}_j^{-1})$

${\mathrm{\Xi }}_j=\mathbf{\Lambda }+{\displaystyle {\sum }_i}E[{\lambda }_{\mathit{\text{ji}}}]{\displaystyle {\sum }_{s,s\prime }}E[x_{\mathit{\text{jis}}}{x}_{\mathit{\text{jis}}\prime }]{\mathrm{\Xi }}_{\text{js}\prime }^t{\mathrm{\Xi }}_{\mathit{\text{js}}}$

${\mathit{\omega ̃}}_j^h=\mathbf{\Lambda }\mathit{h̄}+{\displaystyle {\sum }_{i,s}}E[{\lambda }_{\mathit{\text{ji}}}]E[a_{\mathit{\text{ji}}}]E[x_{\mathit{\text{jis}}}]{\mathrm{\Xi }}_{\mathit{\text{js}}}^t({\mathit{y}}_{\mathit{\text{ji}}}-{\displaystyle {\sum }_d}E[e_{\mathit{\text{jid}}}]{\mathit{f}}_{\mathit{\text{jd}}})$