## 2 Methodology
### Impact Analysis Model
#### 2.3.2 Bayesian Logistic Ordinal Regression  
Compared to traditional logistic regression models, the Bayesian Logistic Ordinal Regression model has three major advantages: first, it is well suited for analyzing the logistic regression relationship between ordered categorical explanatory variables and multiple types of dependent variables; second, Bayesian estimation of the model parameters can handle the uncertainty of the parameters better than maximum likelihood estimation; and third, Bayesian estimation integrates the prior distributions with the likelihood function of the data to derive the posterior distribution of the model parameters, which reflects the probability distribution of the parameters after observing the data and allows for more robust and comprehensive statistical inference. Therefore, this study builds a model for the relationship between cognitive load and other variables of the scene to analyze the effects of various factors on drivers' cognitive load [34; 35]. The utility function of the Bayesian ordered logit regression model is expressed as Equation (3) [36]:  
\[l_{ij}=\mathbf{x}_{ij}^{T}\mathbf{\beta}+\mathbf{b}_{i}+\varepsilon_{ij} \tag{3}\]  
where \(i\) represents the feature variable and \(j\) denotes the number of experimental units for each feature variable. \(\mathbf{x}_{ij}^{T}\) represents the \(i^{*}j\) matrix formed by the feature variables and the number of experiments, which is associated with the fixed effect parameter vector \(\mathbf{\beta}\) of the dimension. \(\mathbf{b}_{i}\) is a random effect and \(\varepsilon_{ij}\) is a logically distributed error term, with its distribution function being calculated in Equation (4):  
\[f\big{(}\varepsilon_{ij}\big{)}=\frac{e^{-\varepsilon_{ij}}}{\left(1+e^{- \varepsilon_{ij}}\right)^{2}} \tag{4}\]Since \(l_{ij}\) is unobservable, it can be indirectly measured through the observable ordered variable \(y_{ij}\). Therefore, \(y_{ij}\) can be defined as Equation (5):  
\[y_{ij}= \begin{cases}1&if\ -\infty<l_{ij}<\gamma_{1},\\ 2&if\ \gamma_{1}<l_{ij}<\gamma_{2},\\ \vdots&\vdots\\ C&if\ \gamma_{C-1}<l_{ij}<\infty\end{cases}  \tag{5}\]  
where \(\gamma_{c}\) represents the boundary between intervals corresponding to the observed outcomes, as shown in Equation (6). \[\gamma^{T}=(\gamma_{min}<\gamma_{1}<\dots<\gamma_{C-1}<\gamma_{max})\text{, with }\gamma_{min}=-\infty\text{, }\gamma_{max}=\infty \tag{6}\]  
In the model, we focus on the cumulative probabilities, such as the probability that \(y\leq\gamma_{c}\). Based on the distribution of \(\varepsilon\), the cumulative response probability for the \(c\) category of the ordinal outcome \(y_{ij}\) is calculated using Equation (7). \[ \begin{split} P\big{(}y_{ij}\leq\gamma_{c}\big{|}\beta,b\big{)}& =\pi_{ij(c)}=P\big{(}l_{ij}\leq\gamma_{c}\big{|}\beta,b\big{)}\\ &=P\Big{(}\mathbf{x}_{ij}^{T}\beta+\mathbf{b}_{i}+\varepsilon_{ij}\leq \gamma_{c}\Big{)}\\ &=P\Big{(}\varepsilon_{ij}\leq\gamma_{c}-\mathbf{x}_{ij}^{T}\beta- \mathbf{b}_{i}\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{)}},\\ & for\ \ c=1,2,\dots,C-1.\end{split}