## 2 Data-Driven Shear Wave Velocity (Vs) Prediction Models
### Bayesian Generalized Linear Model  
The Bayesian framework has been utilized in geotechnical parameter estimations for some time now. This approach offers a powerful paradigm to estimate soil parameters in a probabilistic manner. However, studies on Vs prediction using the Bayesian framework are limited. Bayesian GLM provides a robust framework for integrating CPT data to enhance the accuracy and reliability of Vs predictions. Bayesian GLM are readily available through rstanarm [23] package and can be implemented through R programming language [23]. In this study, we formulate a Bayesian approach to model the relationship between Vsand CPT parameters using a GLM framework. The general form of Bayesian GLM can be expressed as follows [24]:  
Let Y represent Vs and X represent CPT parameters (e.g., cone tip resistance, qc; sleeve friction, fs; friction ratio, Rf). Then, the relationship between Y and predictor variables X can be expressed as  
\[Y=M(X)+E \tag{1}\]  
where Y is predicted Vs, \(M(X)\) is the linear predictor function based on the predictor variables and E is an error term. The linear predictor can be defined as  
\[M(X)=\beta_{0}+\beta_{1}qc+\beta_{2}fs+\ldots+\beta_{p}X_{p} \tag{2}\]  
where \(\beta_{0}\) is the intercept and \(\beta_{i}\) (\(i=1,\ 2,\ldots,p\)) is the coefficient of \(X_{i}\). The Bayesian GLM requires specifying prior distributions for the intercept and coefficients. The values of these parameters are equally likely to be positive or negative, but they are highly unlikely to be far from zero. These beliefs can be represented using normal distributions with a mean of zero and a small standard deviation. The default priors in rstanarm package perform well for most cases. Prior distributions for the coefficients can be defined as  
\[\beta_{i}\sim Normal\left(0,\sigma_{\beta_{i}}^{2}\right) \tag{3}\]  
where \(\sigma_{\beta_{i}}^{2}\) is variance of prior distribution of the coefficients. Similarly, the prior distribution for the intercept can be defined as  
\[\beta_{0}\sim Normal\left(0,\sigma_{\beta_{0}}^{2}\right) \tag{4}\]  
where \(\sigma_{\beta_{0}}^{2}\) is variance of prior distribution of the intercept. The error term is assumed to follow normal distribution with a mean of 0 and variance \(\sigma^{2}\)). \[E\sim Normal\left(0,\ \sigma^{2}\right) \tag{5}\]  
Common choices for the prior distribution of the variance of error include the exponential, inverse gamma, and half-normal distributions. In this study, the exponential prior distribution (the default for GLM in the rstanarm package, with rate parameter \(\lambda=1\)) was considered. It is worth noting that the priors are internally adjusted in rstanarm to account for data scaling and centering. \[\sigma\sim Exponential(\lambda) \tag{6}\]  
Assuming normally distributed errors, the likelihood function can be expressed as  
\[L\left(Y\Big{|}X,\beta,\sigma^{2}\right)=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi \sigma^{2}}}exp\left(-\frac{\left(\ Y_{i}-M(X_{i})\right)^{2}}{2\sigma^{2}}\right) \tag{7}\]  
By using [PERSON] theorem, the posterior distribution of the parameters can be derived as  
\[P(\beta|Y,\ X)\propto L\left(Y\Big{|}X,\beta,\sigma^{2}\right)P(\beta) \tag{8}\]  
The actual values of posterior distributions of the parameters are rarely obtainable through analytical methods. Therefore, Markov Chain Monte Carlo (MCMC) simulation is commonly used to explore the posterior target distribution through random sampling [25; 26]. In MCMC simulations, the Markov chains should converge to stationary distributions. To ensure that the chains mix well within parameter space, visual inspections of the chains and statistical tests, such as the Gelman and Rubin Rhat statistics [27], should be conducted. At convergence, Rhat is expected to be below 1.2, which indicates that the between-chain and within-chain variances for the model parameters are in good agreement [28; 29; 30].