Here we designate a digamma function as
\(\varPsi \left( \cdot \right) \), which will be useful for later discussion, and summarize the property of truncated normal distribution in the probit model.
\(u_{cit}^{(z)} \) follows a normal distribution with mean
\(\varvec{x}_{it}^T \varvec{\beta }_{zi} \) and variance 1. Moreover,
\(u_{cit}^{(z)} \) must satisfy
\(y_{cit} = 1\) if
\(u_{cit} > 0\) and
\(y_{cit} = 0\) if
\(u_{cit} \le 0\). Therefore,
\(u_{cit}^{(z)} \) is generated from a truncated normal distribution as
$$\begin{aligned} u_{cit}^{(z)} \sim \left\{ \begin{array}{ll} TN_{(0,\infty )} \left( {\varvec{x}_{it}^T \varvec{\beta }_{zi}, 1} \right) &{}\quad \text {if }y_{cit} = 1\\ TN_{( - \infty , 0)} \left( {\varvec{x}_{it}^T \varvec{\beta }_{zi}, 1} \right) &{}\quad \text {if }y_{cit} = 0\\ \end{array} \right. \;. \end{aligned}$$
(B5)
Therein,
\(TN_{(n_1, n_2 )} \left( { \cdot , \cdot } \right) \) denotes a normal distribution truncated from
\(n_1 \) to
\(n_2 \). The distribution of
\(u_{cit}^{(z)} \) is therefore expressed as
$$\begin{aligned}&p\left( {u_{cit}^{(z)} \mid \varvec{\beta }_{zi}, z_{cit} ,\varvec{x}_{it}, y_{cit} } \right) \nonumber \\&\quad = \frac{1}{\varOmega _{cit}^{(z)} }\frac{1}{\sqrt{2\pi } }\exp \left\{ { - \frac{1}{2}\left( {u_{cit}^{(z)} - \varvec{x}_{it}^T \varvec{\beta }_{zi} } \right) ^2} \right\} , \end{aligned}$$
(B6)
with
\(\varOmega _{cit}^{(z)} \equiv \left\{ {F\left( { \varvec{x}_{it}^T \varvec{\beta }_{zi} } \right) } \right\} ^{y_{cit}}\left\{ {1-F\left( { \varvec{x}_{it}^T \varvec{\beta }_{zi} } \right) } \right\} ^{\left( {1-y_{cit}} \right) }\). In addition, the expectation value and variance are expressed as
$$\begin{aligned} \varvec{E}\left[ {u_{cit}^{(z)} } \right]&= \varvec{x}_{it}^T \varvec{\varvec{\beta }}_{zi}^*+ \varphi _{cit}^{(z)} \end{aligned}$$
(B7)
$$\begin{aligned} \varvec{V}\left[ {u_{cit}^{(z)} } \right]&= 1 - \varvec{x}_{it}^T \varvec{\beta }_{zi}^*\varphi _{cit}^{(z)} - \left( {\varphi _{cit}^{(z)} } \right) ^2, \end{aligned}$$
(B8)
where
\(\varphi _{cit}^{(z)} \equiv (-1)^{(1-y_{cit})} f\left( { \varvec{x}_{it}^T \varvec{\beta }_{zi}^*} \right) / \varOmega _{cit}^{(z)*} \) and
\(\varOmega _{cit}^{(z) *} \equiv \left\{ {F\left( { \varvec{x}_{it}^T \varvec{\beta }_{zi}^{*} } \right) } \right\} ^{y_{cit}}\left\{ {1-F\left( { \varvec{x}_{it}^T \varvec{\beta }_{zi}^{*} } \right) } \right\} ^{\left( {1-y_{cit}} \right) } \). Consequently, the expected value
\(\varvec{E}_{\ne q_z } \left[ {\log p\left( {\varvec{D},\varvec{\varTheta }} \right) } \right] \) is given as
$$\begin{aligned}&\varvec{E}_{ \ne q_z } \left[ {\log p\left( {\varvec{D},\varvec{\varTheta }} \right) } \right] = \varvec{E}_{q_c } \left[ {\log p\left( {z_{cit} \mid \varvec{\theta }_c } \right) } \right] \nonumber \\&\quad + \varvec{E}_{q_u, q_\beta } \left[ {\log p\left( {u_{cit}^{(z)} \mid \varvec{\beta }_{zi}, z_{cit}, \varvec{x}_{it}, y_{cit} } \right) } \right] + \text {const.} \end{aligned}$$
(B9)
The first term in the right-hand side of Eq. (
B9) is obtained as
\(\varPsi \left( {\gamma _{cz}^*} \right) - \varPsi \left( {\sum \nolimits _{z - 1}^Z {\gamma _{cz}^*} } \right) \) [
6], whereas the second term is evaluated as
$$\begin{aligned}&\varvec{E}_{q_u, q_\beta } \left[ { \log p \left( { u_{cit}^{(z)} \mid \varvec{\beta }_{zi}, z_{cit}, \varvec{x}_{it}, y_{cit} } \right) } \right] \nonumber \\&= \varvec{E}_{q_u, q_\beta } \left[ { - \log \sqrt{2\pi } \varOmega _{cit}^{(z)} - \frac{1}{2} \left( { u_{cit}^{(z)} - \varvec{x}_{it}^T \varvec{\beta }_{zi} } \right) ^2 } \right] \nonumber \\&= -\, \varvec{E}_{q_\beta } \left[ {\log \varOmega _{cit}^{(z)} } \right] - \frac{1}{2}\varvec{E}_{q_u} \left[ {\left( {u_{cit}^{(z)} } \right) ^2} \right] \nonumber \\&\quad + \varvec{E}_{q_u, q_\beta } \left[ {u_{cit}^{(z)} \varvec{x}_{it}^T \varvec{\beta }_{zi} } \right] - \frac{1}{2}\varvec{E}_{q_\beta } \left[ {\left( {\varvec{x}_{it}^T \varvec{\beta }_{zi} } \right) ^2} \right] + \text {const.} \end{aligned}$$
(B10)
To solve Eq. (
B9) for
\(\theta _{citz}^*\), we must evaluate the four terms of Eq. (
B10). The first term includes a CDF from which the expectation value is difficult to obtain analytically. Therefore, we expand the term as a zeroth-order Taylor expansion in terms of the CDF of normal distribution and the logarithm function. Such bold approximation is standard strategies for adapting topic models with VB to practical computation (e.g., zeroth-order Taylor approximation by [
4,
33], and zeroth- and first-order delta approximation by [
8]). The four expectation values in Eq. (
B10) are then written as
$$\begin{aligned} \varvec{E}_{q_\beta } \left[ {\log \varOmega _{cit}^{(z)} } \right]\approx & {} \text {const}, \nonumber \\ \varvec{E}_{q_u} \left[ {\left( {u_{cit}^{(z)} } \right) ^2} \right]= & {} \varvec{V} \left[ {u_{cit}^{(z)} } \right] + \left( {\varvec{x}_{it}^T \varvec{\beta }_{zi}^{*} + \varphi _{cit}^{(z)} } \right) ^2,\nonumber \\ \varvec{E}_{q_u, q_\beta } \left[ {u_{cit}^{(z)} \varvec{x}_{it}^T \varvec{\beta }_{zi} } \right]= & {} \left( {\varvec{x}_{it}^T \varvec{\beta }_{zi}^{*} + \varphi _{cit}^{(z)} } \right) {\varvec{x}_{it}^T \varvec{\beta }_{zi}^*} + \varvec{x}_{it}^T V_{zi}^{\beta *} \varvec{x}_{it}, \nonumber \\ E_{q_\beta } \left[ {\left( {\varvec{x}_{it}^T \varvec{\beta }_{zi} } \right) ^2} \right]= & {} \varvec{x}_{it}^T V_{zi}^{\beta *} \varvec{x}_{it} + \left( {\varvec{x}_{it}^T \varvec{\beta }_{zi}^*} \right) ^2. \end{aligned}$$
(B11)
Finally,
\(\theta _{citz}^*\) is updated as
$$\begin{aligned} \theta _{citz}^*\leftarrow \frac{\exp \left( {\rho _{citz} } \right) }{\sum \limits _{j = 1}^Z {\exp \left( {\rho _{citj} } \right) } }, \end{aligned}$$
(B12)
where
$$\begin{aligned} \rho _{citz} =&\, \varPsi \left( {\gamma _{cz}^*} \right) - \varPsi \left( {\sum \nolimits _{z-1}^Z \gamma _{cz}^*} \right) + \frac{1}{2} \varvec{x}_{it}^T \varvec{\beta }_{zi}^*\varphi _{cit}^{(z)} \nonumber \\&+ \frac{1}{2} \varvec{x}_{it}^T V_{zi}^{\beta *} \varvec{x}_{it} . \end{aligned}$$
(B13)