Algorithm 1 Posterior updates within each iteration.

$$\begin{gathered} \begin{array}{cc}\hfill & \mathrm{Input}:\text{the parameters from the last iteration}\hfill \\ \hfill & \mathrm{Output}:\text{the updated parameters for the next iteration}\hfill \\ \hfill & \\ 1:\text{Draw}{\omega }^{\ast }\sim \mathrm{N}(\omega ,{\tau }_{\omega }^2{I}_q).\text{Update}\omega ={\omega }^{\ast }\text{with probability}\hfill \\ \hfill & \text{min}\left\{1,\frac{\varphi ({\omega }^{\ast }\mid 0,V_{\omega }){\prod }_{i}f(y_i\mid {\omega }^{\ast },•)}{\varphi (\omega \mid 0,V_{\omega }){\prod }_{i}f(y_i\mid \omega ,•)}\right\}.\hfill \\ \hfill & \\ 2:\text{Draw}{\gamma }^{\ast }\sim \mathrm{N}\{\mu (\gamma ),{\tau }_{\gamma }^2{I}_p\},\text{where}\mu (\gamma )=\gamma +\frac{{\tau }_{\gamma }^2}{2}(\frac{\partial \log f}{\partial \gamma }-\frac{1}{2}{\sigma }_{\gamma }^2{\Lambda }_{\gamma }\gamma )\text{with}\frac{\partial \log f}{\partial {\gamma }_j}=\hfill \\ \hfill & \\ {\sum }_{i=1}^n(a^{\prime }({\mathbf{z}}_i^{\mathrm{T}}\omega +{\mathbf{x}}_i^{\mathrm{T}}\beta )+b^{\prime }({\mathbf{z}}_i^{\mathrm{T}}\omega +{\mathbf{x}}_i^{\mathrm{T}}\beta ))x_{ij}\frac{\partial {\beta }_j}{\partial {\gamma }_j}.\text{Update}\gamma ={\gamma }^{\ast }\text{with probability}\hfill \\ \hfill & \text{min}\left\{1,\frac{\varphi (\gamma \mid \mu ({\gamma }^{\ast }),{\tau }_{\gamma }^2{I}_p)\varphi ({\gamma }^{\ast }\mid 0,{\sigma }_{\gamma }^2{\Lambda }_{\gamma }){\prod }_{i}f(y_i\mid {\gamma }^{\ast },•)}{\varphi ({\gamma }^{\ast }\mid \mu (\gamma ),{\tau }_{\gamma }^2{I}_p)\varphi (\gamma \mid 0,{\sigma }_{\gamma }^2{\Lambda }_{\gamma }){\prod }_{i}f(y_i\mid \gamma ,•)}\right\}.\hfill \\ \hfill & \\ 3:\text{Update}\xi =\{j:\mid {\gamma }_j\mid >\lambda \}.\hfill \\ \hfill & \\ 4:\text{For}j\notin \xi ,\text{draw}{\alpha }_j\sim \mathrm{N}(0,{\sigma }_{\alpha }^2).\text{Draw}{\alpha }_{\xi }^{\ast }\sim \mathrm{N}\{\mu ({\alpha }_{\xi }),{\tau }_{\alpha }^2{I}_{\mid \xi \mid }\},\text{where}\mu ({\alpha }_{\xi })=\hfill \\ \hfill & \\ {\alpha }_{\xi }+\frac{{\tau }_{\alpha }^2}{2}(\frac{\partial \log f}{\partial {\alpha }_{\xi }}-\frac{1}{2}{\Sigma }_{\xi }{\alpha }_{\xi })\text{with}\frac{\partial \log f}{\partial {\alpha }_j}={\sum }_{i=1}^n(a^{\prime }({\mathbf{z}}_i^{\mathrm{T}}\omega +{\mathbf{x}}_i^{\mathrm{T}}\beta )+b^{\prime }({\mathbf{z}}_i^{\mathrm{T}}\omega +{\mathbf{x}}_i^{\mathrm{T}}\beta ))x_{ij}\hfill \\ \hfill & \\ \text{for}j\in \xi \text{and}{\Sigma }_{\xi }={\sigma }_{\alpha }^2{I}_{\mid \xi \mid }.\text{Update}{\alpha }_{\xi }={\alpha }_{\xi }^{\ast }\text{with probability}\hfill \\ \hfill & \\ \text{min}\left\{1,\frac{\varphi ({\alpha }_{\xi }\mid \mu ({\alpha }_{\xi }^{\ast }),{\tau }_{\alpha }^2{I}_{\mid \xi \mid })\varphi ({\alpha }_{\xi }^{\ast }\mid 0,{\Sigma }_{\xi }){\prod }_{i}f(y_i\mid {\alpha }_{\xi }^{\ast },•)}{\varphi ({\alpha }_{\xi }^{\ast }\mid \mu ({\alpha }_{\xi }),{\tau }_{\alpha }^2{I}_{\mid \xi \mid })\varphi ({\alpha }_{\xi }\mid 0,{\Sigma }_{\xi }){\prod }_{i}f(y_i\mid {\alpha }_{\xi },•)}\right\}.\hfill \\ \hfill & \\ 5:\text{Draw}{\sigma }_{\gamma }^2\sim \text{IG}({\tilde{a}}_{\gamma },{\tilde{b}}_{\gamma })\text{where}{\tilde{a}}_{\gamma }=a_{\gamma }+\frac{p}{2}\text{and}{\tilde{b}}_{\gamma }=b_{\gamma }+\frac{{\gamma }^{\mathrm{T}}{\Lambda }_{\gamma }^{-1}\gamma }{2}.\hfill \\ \hfill & \\ 6:\text{Draw}{\sigma }_{\alpha }^2\sim \text{IG}({\tilde{a}}_{\alpha },{\tilde{b}}_{\alpha })\text{where}{\tilde{a}}_{\alpha }=a_{\alpha }+\frac{p}{2}\text{and}{\tilde{b}}_{\alpha }=b_{\alpha }+\frac{{\sum }_j{\alpha }_j^2}{2}.\hfill \\ \hfill & \\ 7:(\text{Optional})\text{Draw}{\epsilon }^{\ast }\sim N(\epsilon ,{\tau }_{\epsilon }^2).\text{Update}\epsilon ={\epsilon }^{\ast }\text{with probability}\hfill \\ \hfill & \\ \text{min}\left\{1,\frac{\mid \mathbf{L}+{\epsilon }^{\ast }{\mathbf{I}}_{p_n}{\mid }^{\frac{1}{2}}\epsilon \text{exp}\{-{\epsilon }^{\ast }{\gamma }^T\gamma ∕(2{\sigma }_{\gamma }^2)-(\log {\epsilon }^{\ast }-{\mu }_{\epsilon }{)}^2∕(2{\sigma }_{\epsilon }^2)\}}{\mid \mathbf{L}+\epsilon {\mathbf{I}}_{p_n}{\mid }^{\frac{1}{2}}{\epsilon }^{\ast }\text{exp}\{-\epsilon {\gamma }^T\gamma ∕(2{\sigma }_{\gamma }^2)-(\log \epsilon -{\mu }_{\epsilon }{)}^2∕(2{\sigma }_{\epsilon }^2)\}}\right\}.\hfill \\ \hfill & \\ 8:(\text{Optional})\text{Draw}{\lambda }^{\ast }\sim N_{+}(\lambda ,{\lambda }_l,{\lambda }_u,{\tau }_{\lambda }^2).\text{Update}\lambda ={\lambda }^{\ast }\text{with probability}\hfill \\ \hfill & \\ \text{min}\left\{1,\frac{{\varphi }_{+}(\lambda \mid {\lambda }^{\ast },{\lambda }_l,{\lambda }_u,{\tau }_{\lambda }^2){\prod }_{i}f(y_i\mid {\lambda }^{\ast },•)}{{\varphi }_{+}({\lambda }^{\ast }\mid \lambda ,{\lambda }_{\lambda },{\lambda }_u,{\tau }_{\lambda }^2){\prod }_{i}f(y_i\mid \lambda ,•)}\right\}.\hfill \end{array} \end{gathered}$$