## 3 Proposed Methods Based on Causal Inference  
### Mathematical Representation of Causal Network  
This section provides a detailed description of the feature selection method in data mining based on a causal network. The dataset W consists of two parts: the input features and the output results. Let \(X=\left\{x_{ij}\right\}\) denote the input matrix and \(Y=\left\{y_{i}\right\}\) represent the results, where \(i=1,\ 2,\ \dots,\ n,\ j=1,\ 2,\ \dots,\ m\). The equation below illustrates this relationship. \[W=\left[X,Y\right]= \begin{pmatrix}x_{11}&x_{12}&\dots&x_{1m}&y_{1}\\ x_{21}&x_{22}&\dots&x_{2m}&y_{2}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ x_{n1}&x_{n2}&\dots&x_{nm}&y_{n}\\ \end{pmatrix}  \tag{1}\]  
The algorithm employed in this study introduces a novel approach for describing Bayesian networks. By solving an optimization problem with smooth constraints, we can determine the optimal directed acyclic graph. Specifically, for a given dataset \(W=\left(\omega_{ij}\right)\in R^{d\times d}\), where \(A(W)\in\left\{0,1\right\}^{d\times d}\) is a binary adjacency matrix representing a directed graph, the description is as follows: \(\left[A(W)\right]_{ij}=1\Leftrightarrow\omega_{ij}\
eq 0\); otherwise, it is 0. Moreover, let \(D\subset\left\{0,\ 1\right\}^{d\times d}\) represent a subset of the binary matrix \(B\), where \(B\) is the adjacency matrix of the acyclic graph. Consequently, the optimization problem for Bayesian structure is transformed into the following nonconvex form:  
\[min_{W\in R}^{d\times d}Q(W;X)\text{ {\em subject to }}h(W)=0 \tag{2}\]  
The loss function \(Q\), which is associated with the data, is denoted by \(X\in R^{n\times d}\), representing the data matrix. The smoothing function \(h:R^{d\times d}\to R\) is only applicable when \(A(W)\in D\) and \(h(W)=0\). Consequently, the Bayesian network's structure, based on graph theory, is transformed into a nonconvex optimization problem. This problem can be solved using mathematical optimization tools. However, prior to finding a formal solution, it is necessary to further characterize and express the acyclic constraint in the aforementioned equation using mathematical methods. This step facilitates the subsequent optimization problem's solution. This section presents a novel representation method for acyclic constraints, utilizing the concept of matrix indices to facilitate subsequent optimization problems. The matrix exponent, analogous to the exponential function, is introduced as a function applicable to square matrices. In this representation method, the variables of the exponential function are replaced by a block matrix. The specific definition is as follows:  
\[e^{B}=\sum\
olimits_{k=0}^{\infty}\frac{1}{k!}B^{k} \tag{3}\]  
Notably, \(B\) is an \(n\times n\) matrix if and only if \(\mathit{trexp}(B)=d\), where \(B\in\{0,1\}^{d\times d}\) represents the adjacency matrix of a DAG. Proof. : If \(\left(B^{k}\right)_{ii}=0\) holds for all \(k\geq 1\) and all \(i\), \(B\) indicates the absence of loops in the directed graph, resulting in:  
\[\mathit{trexp}(B)=\mathit{tr}\sum\limits_{k=0}^{\infty}\frac{B^{k}}{k!}= \mathit{tr}\sum\limits_{k=1}^{\infty}\frac{B^{k}}{k!}+d=\mathit{tr}\sum \limits_{k=1}^{\infty}\sum\limits_{i=1}^{d}\frac{(B^{k})_{ii}}{k!}+d=d \tag{4}\]  
Given the usage of the block matrix \(B\) in the aforementioned theorem, and since \(B\) is a binary matrix, it is not generally applicable to diverse data types. Therefore, a method is required to substitute \(B\) with an arbitrary weight matrix \(W\). Here, we employ the Hadamard product of matrices, defined as follows:  
\[A\circ B= \begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{m1}&a_{m2}&\ldots&a_{mn}\end{pmatrix} \circ \begin{pmatrix}b_{11}&b_{12}& \ldots&b_{1n}\\ b_{21}&b_{22}&\ldots&b_{2n}\\ \vdots&\vdots&&\vdots\\ b_{m1}&b_{m2}&\ldots&b_{mn}\end{pmatrix} =