## 2 Methodology
### Impact Analysis Model
#### 2.3.2 Bayesian Logistic Ordinal Regression
 \tag{7}\]  
This can be formulated as a cumulative logit model included in Equation (8):  
\[log\Bigg{(}\frac{\pi_{ij(c)}}{1-\pi_{ij(c)}}\Bigg{)}=\gamma_{c}-\ \mathbf{x}_{ij}^{T}\beta-\mathbf{b}_{i},\ \ \ for\ c=1,2,\dots,C-1. \tag{8}\]  
Using the inverse link for this model, we can calculate \(P(y=c|\beta,b)=\pi_{ij(c)}\) in Equation (9):  
\[\pi_{ij(c)}=P\big{(}\gamma_{c-1}<l_{ij}<\gamma_{c}\big{)}=\frac{exp\Big{(} \gamma_{c}-\mathbf{x}_{ij}^{T}\beta-\mathbf{b}_{i}\Big{)}}{1+exp\Big{(}\gamma_{c}- \mathbf{x}_{ij}^{T}\beta-\mathbf{b}_{i}\Big{)}}-\frac{exp\Big{(}\gamma_{c-1}-\mathbf{x}_{ ij}^{T}\beta-\mathbf{b}_{i}\Big{)}}{1+exp\Big{(}\gamma_{c}-\mathbf{x}_{ij}^{T}\beta-\mathbf{b}_{i} \Big{)}} \tag{9}\]  
Then, the joint posterior density of the parameter vector and latent variable becomes Equation (10):  
\[P\Big{(}\beta,\gamma,b,\sigma_{\beta}^{2},\sigma_{b}^{2},l\ /\ \mathbf{y}\Big{)}\propto\ P(\mathbf{y}/\ I,\gamma)\ P(\ I/\ /\ \beta,b)\ P(\gamma)\times P\Big{(}\beta\ /\sigma_{\beta}^{2}\Big{)}\ P \Big{(}\ \mathbf{b}\ /\sigma_{\beta}^{2}\Big{)}\ P\Big{(}\sigma_{\beta}^{2}\Big{)}\ P \Big{(}\sigma_{\beta}^{2}\Big{)} \tag{10}\]  
The prior for the \(C-1\) unknown thresholds has been given as order statistics from \(U(\gamma_{min},\gamma_{max})\) distribution using Equation (11) [37]:  
\[P(\gamma)=(C-1)!\bigg{(}\frac{1}{\gamma_{max}-\gamma_{min}}\bigg{)}^{C-1}I( \gamma\in T) \tag{11}\]  
where \(T=\{(\ \gamma_{1},\dots,\gamma_{max})|\gamma_{min}<\gamma_{1}<\dots<\gamma_{C-1}< \gamma_{max}\}\). The posterior conditional distribution of \(\beta\) is defined by Equation (12)  
\[\beta|ELSE\ \sim N_{p}\Big{(}\tilde{\beta}_{0},\tilde{\Sigma}_{0}\Big{)} \tag{12}\]where  
\[\widetilde{\mathbf{\Sigma}}_{0}=\Big{(}\mathbf{\Sigma}_{0}^{-1}\sigma_{\beta}^{-2}+\mathbf{X }^{T}\mathbf{D}_{\omega}\mathbf{X})^{-1},\]  
\[\widetilde{\mathbf{\beta}}_{0}=\widetilde{\mathbf{\Sigma}}_{0}\Big{(}\mathbf{\Sigma}_{0}^{-1 }\sigma_{\beta}^{-2}\mathbf{\beta}_{0}-\mathbf{X}^{T}\mathbf{D}_{\omega}Zb+\mathbf{X}^{T}\mathbf{D }_{\omega}l).\]  
with  
\[l=\Big{[}l_{1}^{T},\ldots,l_{l}^{T}]^{T},l_{i}=\Big{[}l_{i_{1}}^{T},\ldots,l_{ i_{m_{i}}}^{T}]^{T},\]  
\[X=\Big{[}X_{1}^{T},\ldots,X_{l}^{T}]^{T},X_{i}=\Big{[}x_{i1},\ldots,x_{in_{i} }]^{T}\]  
\[Z= \begin{bmatrix}I_{n_{1}}&\mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{0}&I_{n_{2}}&\cdots&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{0}&\mathbf{0}&\cdots&I_{n_{1}}\end{bmatrix} ,\mathbf{D}_{\omega}=diag(\mathbf{D}_{ \omega 1},\ldots,\mathbf{D}_{\omega 1})\]  
\[\mathbf{D}_{\omega}=diag(\mathbf{D}_{\omega 1},\ldots,\mathbf{D}_{\omega 1}),\]  
\[\mathbf{D}_{\omega i}=diag\big{(}\omega_{i1},\ldots\omega_{in_{i}}).\]  
Equation (12) indicates that, given the data (denoted as \(ELSE\)), the posterior distribution of the regression coefficient \(\mathbf{\beta}\) is a multivariate normal distribution with mean \(\widetilde{\mathbf{\beta}}_{0}\). The mean \(\widetilde{\mathbf{\beta}}_{0}\) and the covariance matrix \(\widetilde{\mathbf{\Sigma}}_{0}\) are determined jointly by the prior covariance, observed data, and corresponding weighting information. This allows the estimates of the regression coefficients to be dynamically adjusted based on the observed data, providing a more accurate reflection of the model's response to the data.