## 1 Introduction  
The composition and physical and mechanical properties of clayey soil are important indexes in geotechnical engineering, among which the undrained shear strength index of clay is the most important index for calculation of the bearing capacity of foundation soil, calculation of the soil pressure of the foundation pit, and analysis of slope stability. Cross plate shear tests in the field, direct shear tests in the laboratory, triaxial compression tests, and other such direct measurement methods are time-consuming and expensive [1]. To evaluate through indirect methods, theoretical derivation and empirical methods are effective methods. Based on many experiments, [PERSON] and [PERSON] et al., respectively, proposed empirical equations for the undrained shear strength of clay [2; 3]. [PERSON] et al. simulated the uncertainty of soil parameters and proposed an effective non-stationary random field model for undrained shear strength parameters [4]. Indirect methods also include measurements obtained from dilatometer tests (DMT) based on empirical equations [5; 6]. The application of machine learning in the field of geotechnical engineering has gradually become a research hotspot, including slope stability testing, TBM performance evaluation, rockburst vibration estimation, and pile foundation evaluation, etc., and has been proposed, expanded upon and applied by many researchers [7; 8; 9; 10]. The emergence of artificial intelligence techniques holds great potential for solving soil parameter estimation problems involving complex soil-structure interactions [11; 12; 13]. However, compared with other directions in the field of geotechnical engineering, research devoted to soil parameter estimation is still very limited. The CatBoost algorithm can effectively solve gradient offset and prediction offset problems, improving the accuracy and generalization ability of the calculation [14]. Therefore, it has been widely used in crop evapotranspiration estimation in hydrology, data-driven seismic performance evaluation of corroded RC columns in civil engineering, and groundwater salinity prediction in multi-layer coastal aquifers in geotechnical engineering [14; 15; 16; 17; 18]. Moreover, the combination of Bayesian optimization with machine learning algorithms is still relatively rare, and [PERSON] et al. use extreme gradient boosting and random forest based on Bayesian optimization in geotechnical engineering to effectively capture the relationship between undrained shear strength and various fundamental soil parameters [19]. [PERSON] et al. used the good performance of Bayesian, functional, and meta-ensemble machine learning models to generate land subsidence susceptibility (LSS) maps [20]. The Cambridge model and modified Cambridge model are two elastic-plastic constitutive models commonly used in geotechnical engineering to describe the shear properties of soils [21]. They are typical of the constitutive relationships of soils under undrained conditions. The Cambridge model describes the shear properties of soils under circular stress paths, while the modified Cambridge model modifies the stress paths in the plastic potential function to better describe the shear properties of soils under non-circular stress paths [22]. Under the Bayesian framework, the CatBoost algorithm (CatBoost-Bayesian) based on a Bayesian optimization algorithm was developed to obtain the feature importance level of soil parameters affecting the undrained shear strength of clay by using the experimental samples of Finnish clay, so as to adaptively couple the theoretical equation of undrained shear strength of \(K_{0}\) consolidated clay, which was derived from the modified Cambridge model. Then, the theoretical equation of undrained shear strength of isotropically consolidated clay was established from the critical state of clay parameters. Finally, this paper analyzed the uncertainty of the parameters of the equation and verified the rationality of the calculation results. The rest of this study was arranged as follows.