## 2 Materials and Methods
### Bayesian Optimization Algorithm (SMAC) and k-Fold Cross-Validation  
Hyperparameter optimization is a combinatorial optimization problem that cannot be optimized by gradient descent as general parameters. Evaluating a set of hyperparameter problems often involves issues such as resource allocation and efficiency. The simpler hyperparameter optimization methods include manual search, grid search, and random search. In this study, we used the currently popular adaptive hyperparameter search method of Bayesian optimization [24; 25; 26]. Based on the combinations of hyperparameters that were already tested, this method can predict the next combination that is more likely to bring the greatest benefit. This study is implemented using the optuna automatic hyperparameter optimization framework.  
SMBO stands for sequential model-based optimization. The so-called serialization refers to the optimization by iterative method one trial at a time. SMBO is a specific implementation form of Bayesian optimization.  
SMAC, proposed by [PERSON] et al. [27], stands for sequential model-based optimization for general algorithm configuration; the model originates from the random forest model. The random forest (RF) algorithm is a combination of the Bootstrap Aggregating algorithm proposed by [PERSON] in 1996 and the stochastic subspace algorithm proposed by [PERSON] [28] in 1998. SMAC was initially designed to remove some of the limitations of SMBO to make it applicable to general algorithm configuration problems with many classification parameters and benchmark sets of instances, and to solve the case in which the parameter type cannot be discrete in the Gaussian regression process.  
During the process of model training, the problem of data overfitting often occurs. The model can match the training data well, but cannot predict the data outside the training set well. This affects the accuracy of the final evaluation results.  
K-fold cross-validation (CV) is widely used as a criterion for model selection [29], and was originally created to reduce the computational cost of leave-one-out cross-validation (LOOCV) [30]. It divides a dataset into K equal samples, of which K-1 samples are used to build the model for training, and the remaining 1 sample is used for validation. During this K-time iteration, each part of K is successively assigned as validation data.  
[PERSON] [31] proposed a new method of choosing K-fold to ensure K = ln(\(n\)) and guarantee \(n/\text{K}>3d\) simultaneously. After verifying the feasibility of it, where n denotes a data point and d denotes the number of parameters (in this study, \(n\) = 202, \(d\) = 12, K = 5.3), and combining it with [PERSON] [32] method, K was setted to 5. Therefore, this study used a five-fold cross-validation method for validation (Figure 2).  
Figure 2: Five-fold cross-validation.  
To reduce the reliance on empirical rules and inefficient brute force search, Bayesian optimization methods are applied to determine the appropriate model hyperparameters for the CatBoost algorithm (CatBoost-Bayesian). With appropriate model hyperparameters, the CatBoost-Bayesian hybrid model is able to provide a data-driven ranking of clay feature importance and properly uncover the intrinsic linkage of soil parameter essences; however, it cannot give full interpretability. To address this issue, the clay feature importance is adaptively coupled with the theoretical equations derived from the modified Cambridge model to estimate the undrained shear strength of isotropically consolidated clay under a Bayesian probabilistic framework, taking into account the parameter and model uncertainties, which are explained in detail in Sections 2.3 and 2.4.