## 2 Related Work
### Gpr  
Due to the numerous excellent properties of Gaussian distribution, GPR offers good adaptability when dealing with small sample sizes, high dimensions, nonlinearity and other problems [27]. In recent years, GPR has become a trending topic in the field of machine learning [28; 29; 30]. There are two different ways to derive GPR, which can be explained by weight space and function space. Since the derivation of weight space is deeply dependent on Bayesian theory, this paper specifically explains it from the perspective of weight space for a better understanding. Let us start with the simplest linear regression model with noise. The formula is shown as follows:  
\[f(\mathbf{x})=\mathbf{x}^{T}\mathbf{w},\mathbf{y}=f(\mathbf{x})+\mathbf{\varepsilon} \tag{4}\]  
where \(\mathbf{x}=\big{(}x_{1},x_{2},\dots,x_{p}\big{)}^{T}\) is the input of dimension \(p\), \(\mathbf{w}=\big{(}w_{1},w_{2},\dots,w_{p}\big{)}^{T}\) is the weight vector of dimension \(p\), and \(y\) represents the true value. Here, we assume that the noise \(\mathbf{\varepsilon}\) obeys a Gaussian distribution, with \(0\) as the mean and \(\sigma_{n}^{2}\) as the variance. Since we have previous knowledge about the weight vector, it is assumed that it obeys a Gaussian distribution with a mean value of \(0\) and variance of \(\sum_{p}\), namely, \(\mathbf{\varepsilon}\sim N\big{(}0,\sigma_{n}^{2}\big{)}\), \(\mathbf{w}\sim N\big{(}0,\sum_{p}\big{)}\). Through a series of observed training data \(\mathbf{D}=\big{(}\mathbf{X}\mathbf{y}\big{)}=\big{(}\mathbf{\chi}_{i}\mathbf{y}_{i}\big{)}\,|\, \mathbf{x}_{i}\mathbf{\in}R^{n},\,y_{i}\in R,I=1,\dots\,\mu\big{)}\) and the assumption of independent noise distribution, we can calculate the likelihood function as follows:  
\[ \begin{split}& p(\mathbf{y}|\mathbf{x},\mathbf{w})=\prod\limits_{i=1}^{n}p(y_{ i}|\mathbf{x}_{i},\mathbf{w})=\frac{1}{\big{(}2\pi\sigma_{n}^{2}\big{)}^{n/2}}\exp \Bigl{(}-\frac{1}{2\sigma_{n}^{2}}\big{(}\mathbf{y}-\mathbf{X}^{T}\mathbf{w}\big{)}^{T} \big{(}\mathbf{y}-\mathbf{X}^{T}\mathbf{w}\big{)}\Bigr{)}\\ &\sim N\big{(}\mathbf{X}^{T}\mathbf{w},\sigma_{n}^{2}\mathbf{I}_{n}\big{)} \end{split}  \tag{5}\]  
With the likelihood function and the prior distribution of weights, the posterior distribution of weights can be obtained using a Bayesian formula. First, according to [PERSON] formula:  
\[p(\mathbf{w}\,|\mathbf{D})=\frac{p(\mathbf{y}|\mathbf{X},\mathbf{w})\,p(\mathbf{w})}{p(\mathbf{y}\,|\mathbf{X} )}=\frac{p(\mathbf{y}|\mathbf{X},\mathbf{w})\,p(\mathbf{w})}{\int p(\mathbf{y}|\mathbf{X},\mathbf{w})\,p( \mathbf{w})d\mathbf{w}}\propto p(\mathbf{y}|\mathbf{X},\mathbf{w})\,p(\mathbf{w}) \tag{6}\]  
In the formula, \(\propto\) represents the ignored constant scaling coefficient. In mathematics,where \(\mathbf{\mu}_{w}=\sigma_{n}^{-2}\left(\sigma_{n}^{-2}\mathbf{X}\mathbf{X}^{\mathrm{T}}+\Sigma_{ p}^{-1}\right)^{-1}\mathbf{X}\mathbf{y}\), \(\Sigma_{w}=\left(\sigma_{n}^{-2}\mathbf{X}\mathbf{X}^{\mathrm{T}}+\Sigma_{p}^{-1}\right) ^{-1}\). According to the form of the Gaussian density function, we can easily determine that it is a multidimensional Gaussian distribution with a mean value and covariance matrix, namely:  
\[p(\mathbf{w}|\mathbf{D})\,\sim N(\mathbf{\mu}_{w},\Sigma_{w}) \tag{8}\]  
With the posterior distribution of this weight, we can predict a group of test points \(\mathbf{z}\). Here, we need to use the concept of total probability. \(p(f_{*}|\mathbf{z},\mathbf{D})\) is the integral of all possible parameters and their corresponding posterior distribution, that is, the predicted distribution of the predicted value \(f_{*}\).