# Research on Slope Stability Based on Bayesian Gaussian Mixture Model and Random Reduction Method
## 1 Introduction  
Slope stability has long been a critical concern in geotechnical engineering [1]. Unstable slopes may trigger landslides, collapses, and other geological hazards, posing severe threats to mining safety and economics. Analyzing slope stability is therefore of significant importance [2]. Currently, the primary metric for evaluating slope stability is the safety factor [3], which represents the safety margin and is typically derived through the strength reduction method combined with numerical simulation. Traditional strength reduction methods [4] proportionally reduce cohesion (\(c\)) and internal friction angle (\(\varphi\)) until the slope reaches its limit state, using the reduction coefficient as the safety factor. As early as 2002, [PERSON] et al. [5] combined finiteelement methods with strength reduction to calculate slope safety factors. Subsequently, [PERSON] et al. [6] compared three numerical criteria for slope instability--numerical non-convergence, sudden displacement changes at characteristic points, and generalized plastic strain penetration--and proposed using computational convergence as the primary criterion. [PERSON] et al. [7] introduced residual displacement increments based on characteristic point displacement mutation criteria to eliminate subjective errors in manual judgment of displacement mutations. Beyond numerical methods, machine learning has been applied to predict slope stability. [PERSON], [PERSON] et al. [8] utilized multiple classification algorithms to predict slope stability under different joint types (parallel deterministic, cross-jointed, Baecher, Veneziano, and Voronoi). [PERSON], [PERSON] et al. [9] employed the ARIMA model to forecast slope failure timing. Furthermore, considering the inherent uncertainties and variability in geotechnical parameters, the traditional deterministic safety factor framework exhibits certain limitations. Consequently, probabilistic methods have gained significant attention in slope stability assessment. For instance, [PERSON] et al. [10]. comprehensively reviewed currently available probabilistic approaches, providing detailed descriptions of various methodologies whilst elucidating the efficiency and drawbacks of each technique and their evolution in handling probabilistic slope analysis. Although these methods hold considerable potential in slope stability research, the field remains nascent and requires further refinement. Consequently, quantitatively determined deterministic safety factors typically remain the preferred metric for engineering practice. As research on rock mass failure mechanisms deepens, studies reveal that the decay rates of \(c\) and \(\varphi\) during progressive failure are inconsistent [11]. Traditional proportional reduction assumptions contradict this mechanism, prompting the development of dual-coefficient reduction methods. [PERSON] et al. [12] established curves for critical slopes under varying geometries and rock densities, proposing a dual-parameter reduction method based on the shortest path to the critical curve. However, this method assumes specific reduction paths and lacks theoretical justification for its composite safety factor. [PERSON] et al. [13] derived reduction ratios by assuming linear decay between peak and residual strength parameters, defining safety factors via reference slope probabilities. Although physically meaningful, this approach relies on linear decay assumptions. [PERSON] et al. [14] proposed a dual-coefficient method weighted by energy dissipation contributions during parameter reduction, yet it still presumes reduction paths. However, the approach of integrating energy evolution in this paper is noteworthy. In summary, whilst current research addresses non-proportional reduction \(c\) of and \(\varphi\), critical gaps remain as composite safety factors lack physical meaning and reduction paths are often arbitrarily assumed.