## 5 Modeling Results and Discussion  
### Full Bayesian Estimation  
A Full Bayesian (FB) approach has been widely recommended for determining the unknown parameters of a crash model [4; 11]. The FB approach treats unknown parameters as random variables, and the corresponding distribution characteristics are defined by prior distribution. The core is to obtain posterior distribution by combining prior distribution and traditional likelihood function [43]. Hence, it is unavoidable to specify the prior distribution of (hyper-) parameters. In this study, a non-informative prior distribution was adopted due to lack of relevant knowledge. Specifically, a diffused normal distribution was specified for fixed parameters \((\beta_{l},\ \beta_{0})\) and random parameters \((\beta_{i,l},\ \beta_{i,0})\), which can be expressed as:  
\[\left\{ \begin{array}{l}\beta_{l}\text{$\sim$ normal(0_{\tau},\ $10^{4}I_{\tau})$}\\ \beta_{0}\text{$\sim$ normal(0_{\tau},\ $10^{4}I_{\tau})$}\end{array} \right., \left\{ \begin{array}{l}\beta_{i,l}\text{$\sim$ normal(0_{\tau},\ $10^{4}I_{\tau})$}\\ \beta_{i,0}\text{$\sim$ normal(0_{\tau},\ $10^{4}I_{\tau})$}\end{array} \right., \tag{20}\]  
where \(0_{\tau}\) denotes a zero vector of the form \(\tau\times 1\). \(I_{\tau}\) denotes a unit matrix of the form \(\tau\times\tau\). Meanwhile, the prior distribution of precision parameters (\(\varphi_{k},\ \sigma_{k0}^{2}\), and \(\sigma_{u_{k}}^{2}\)) and \(\sigma_{s_{k}}^{2}\) adopted the inverse gamma distribution and gamma distribution, respectively, which can be expressed as:  
\[\left\{ \begin{array}{l}\varphi_{k}\text{$\sim$ inverse gamma(0.001,\ 0.001)}\\ \sigma_{k,0}^{2}\text{$\sim$ inverse gamma(0.001,\ 0.001)}\\ \sigma_{u_{k}}^{2}\text{$\sim$ inverse gamma(0.001,\ 0.001)}\\ \sigma_{s_{k}}^{2}\text{$\sim$ gamma(1+\sum\frac{v_{w}}{2},\ 1+\frac{u}{2})$}\end{array} \right., \tag{21}\]  
where 0.001 indicates the parameter of the inverse gamma distribution. \(v_{w}\) represents the term contributed by each road segment, and the corresponding calculation rule is exhibited as follows [11]:  
\[v_{w}=\left(\sum_{\alpha\
eq\ \omega^{\prime}}\delta_{w,\ \omega^{\prime}} \right)s_{k}^{\text{IP}}\left(s_{k}^{w}-\overline{s_{k}^{w}}\right). \tag{22}\]  
The computation of high-dimensional integrals in FB inference is difficult. Hence, a MCMC (Markov Chain Monte Carlo) simulation technique based on Metropolis-Hastings sampling was employed. The MCMC technique generates samples from posterior distribution and provides an efficient way to estimate the FB model. Considering the high complexity, 3 parallel MCMC simulation chains were created, and 30,000 iterations were maintained for each chain. The mean value of prior distribution was used as the initial value of model parameter and the real value was obtained by sufficient sampling simulation and removal of the burn-in period [11]. The first 15,000 iterations were discarded as the burn-in period because the posterior distribution achieves convergence after that; the other iterations were used for model estimation. Meanwhile, an intermittent sampling method was adopted to reduce sample correlation--that is, 1 of every 10 samples was reserved. The model estimation was implemented through the free software WinBUGS. Several test measures were performed to determine the convergence of the MCMC chain and the relevance of the samples. First, the MCMC dynamic plot traces were visually checked. If parameter values lie within a region without strong periodicities, the conclusion of MCMC chain convergence can be drawn. Second, if all Brooks-Gelman-Rubin (BGR) statistics were lower than 1.2, the MCMC chains are considered to be convergent. The conclusion of model convergence can be obtained when both diagnostics are satisfied at the same time [64]. In addition, the autocorrelation plots (autocorrelation functions) were carefully examined to determine that the iterative chains were adequately close to independent and identically distributed (IID).