# Research on Slope Stability Based on Bayesian Gaussian Mixture Model and Random Reduction Method
## 2 Construction of Slope Instability Criteria Using Bayesian Gaussian Mixture Model
### Bayesian Estimation Algorithm  
The likelihood equation of the Gaussian Mixture Model typically lacks an analytical solution. Combining the Bayesian algorithm for iterative computation is one method for solving the Gaussian Mixture Model. Bayesian estimation [24], as a statistical inference method based on [PERSON] theorem, is fundamentally different from the traditional frequentist approach. In the Bayesian estimation framework, the unknown parameters of the model are treated as random variables, unlike the traditional frequentist approach, which considers them as fixed but unknown constants. The key point lies in accurately calculating the posterior distribution of the parameters based on prior knowledge and observed data. Here, the prior distribution carries the accumulated knowledge and assumptions about the parameters before observing the data. Subsequently, using [PERSON] formula the prior distribution and the likelihood function of the observed data are integrated to derive the posterior distribution. Finally, parameter estimation and inference are conducted based on the obtained posterior distribution. The specific steps for estimating the parameters of the Gaussian Mixture Model using the Bayesian algorithm are as follows.  
In the Gaussian Mixture Model, the parameters to be estimated are the mixing coefficient \(\pi_{k}\), the mean vector \(\mu_{k}\)  
where the mixing coefficient follows a Dirichlet distribution, \(\pi_{k}\sim Dir(\alpha)\), where \(\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{k})\) is the parameter vector of the Dirichlet distribution; the mean vector follows a normal distribution, \(\mu_{k}\sim\mathcal{N}(m_{0},V_{0})\), where \(m_{0}\) is the prior mean and \(V_{0}\) is the prior covariance; the covariance matrix follows an inverse Wishart distribution, \(\sum_{k}\ \sim IW(v_{0},S_{0})\), where \(v_{0}\) and \(S_{0}\) are the parameters of the inverse Wishart distribution; and \(d\) is the data dimensionality.  
Assuming a set of observed vectors as \(X=[x_{1},x_{2},\ldots,x_{n}]^{\mathsf{T}}\), and letting the unknown parameter be \(\theta\), the posterior probability of \(\theta\) can be expressed as follows:  
\[p(\theta|X)=\frac{p(X|\theta)p(\theta)}{p(X)} \tag{16}\]  
where \(p(\theta|X)\) is the posterior probability of \(\theta\); \(p(X|\theta)\) is the probability density function of \(X\); \(p(\theta)\) is the prior estimate of \(\theta\); and \(p(X)\) is the observed probability of \(X\).  
Under the given observed event \(\theta\), \(p(X)\) is a measurable constant, and \(p(X|\theta)\) can be considered the likelihood function of event \(\theta\). Thus, the posterior probability \(p(\theta|X)\) is proportional to \(p(X|\theta)p(\theta)\), which is shown as follows:  
\[p(\theta|X)\propto p(X|\theta)p(\theta) \tag{17}\]  
From Equation (17), the posterior distribution of the mixing coefficient \(\pi_{k}\) is \(\pi_{k}\sim Dir(\alpha^{*})\), where \(\alpha*=(\alpha_{1}+n_{1},\alpha_{2}+n_{2},\ldots,\alpha_{k}+n_{k})\), and \(n_{k}\) is the number of samples belonging to the k-th Gaussian component. The posterior distribution of the mean vector \(\mu_{k}\) is \(\mu_{k}\sim\mathcal{N}(m^{*},V^{*})\) where \(V^{*}=(nkV_{0}^{-1}+V_{0}^{-1})^{-1}\) and \(m^{*}=\frac{V^{*}(n_{k}V_{0}^{-1}v_{0}^{-1}n_{0})}{n_{k}V_{0}^{-1}+V^{*}}\). The posterior distribution of the covariance matrix \(\sum_{k}\ \text{is}\ \sum_{k}\ \sim IW(v^{*},S^{*})\), where \(v^{*}=v_{0}+n_{k}\) and \(S^{*}=S_{0}+S_{k}\).  
By calculating the expected value of the posterior distribution, the parameter estimates can be obtained. The specific principle of the Bayesian estimation algorithm for computing the parameter values of the Gaussian Mixture Model is illustrated in Figure 1: