Table 1. All partial derivatives of ℒ_q,θ.

T_A is a function of a_ij and z_j. For details see S1 Notes.

$\frac{\partial {\mathcal{L}}_{q,\theta }}{\partial {\mu }_i}$	$\frac{1}{{\sigma }_0^2}{\sum }_{j=1}^M{a}_{ij}\left(z_j-{\sum }_{k=1}^M{a}_{kj}{\mu }_k\right)-\frac{(1-q_i){\mu }_i}{{\delta }^2}-\frac{\left(q_i\right){\mu }_i}{{\sigma }_{\beta }^2}$
$\frac{\partial {\mathcal{L}}_{q,\theta }}{\partial {\sigma }_i^2}$	$\frac{-1}{4{\sigma }_0^2}{\sum }_{j=1}^{M}2a_{ij}^2-\frac{1}{2}\left(\frac{(1-q_i)}{{\delta }^2}+\frac{\left(q_i\right)}{{\sigma }_{\beta }^2}-\frac{1}{{\sigma }_i^2}\right)$
$\frac{\partial {\mathcal{L}}_{q,\theta }}{\partial q_i}$	$‐\left(\log \right(\frac{{\sigma }_{\beta }}{\delta })-\frac{{\sigma }_i^2+{\mu }_i^2}{2{\delta }^2}+\frac{{\sigma }_i^2+{\mu }_i^2}{2{\sigma }_{\beta }^2}+\log \frac{q_i}{{\pi }_1}-\log \frac{1-q_i}{1-{\pi }_1})$
$\frac{\partial {\mathcal{L}}_{q,\theta }}{\partial {\pi }_1}$	${\displaystyle \sum _{i=1}^M}\frac{{\pi }_1-q_i}{{\pi }_1-{\pi }_1^2}$
$\frac{\partial {\mathcal{L}}_{q,\theta }}{\partial {\sigma }_{\beta }^2}$	${\displaystyle \sum _{i=1}^M}\frac{q_i}{2{\sigma }_{\beta }^4}\left({\sigma }_i^2+{\mu }_i^2-{\sigma }_{\beta }^2\right)$
$\frac{\partial {\mathcal{L}}_{q,\theta }}{\partial {\sigma }_0^2}$	$\frac{T_A-M{\sigma }_0^2}{2{\sigma }_0^4}$