## 3 Proposed Method
### Heuristic Causal Inference  
With the global and adjacent local causal networks obtained above, we design a heuristic causal inference method to quantitatively calculate the causal pathway contribution degree of cause variables to the target variable. In general, once a global causal network among variables is learned, the causal effects between variables can be calculated by using various graph search algorithms combined with quantitative causal inference methods. However, the calculation process is often very complex and is not feasible for large, dense causal networks. In view of this, a heuristic strategy for approximate causal inference is proposed. The basic idea of our method comes from a perceptual understanding of the physical structure of a causal network. In general, it can be inferred that the more causal pathways via a cause to the target, the greater the causal effect of the cause on the target. Under the same conditions, it can also be inferred that the shorter the causal pathway length of a cause variable from the target variable, the greater the causal effect of the cause on the target. Based on the basic understanding of the above two aspects, the framework of the heuristic causal inference model is shown in Figure 2.  
For the convenience of expression, we provide the following definitions:  
**Definition 3**. : _Global Causal Pathway--In the global causal network, a variable \(x_{i}(i=1,2,\cdots,m)\) is the direct or indirect cause of the target variable \(t\), and the causal pathway \((\cdots\to x_{i}\to\cdots\to t)\) that points to the target variable \(t\) through \(x_{i}\) is defined as the global causal pathway from \(x_{i}\) to \(t\), as shown in Figure 3\(a\)._  
**Definition 4**. : _Local Causal Pathway--In the global causal network, a variable \(x_{i}(i=1,2,\cdots,m)\) is the direct or indirect cause of the target variable \(t\), and the causal pathway \((x_{i}\to\cdots\to t)\) that points to the target variable \(t\) from \(x_{i}\) is defined as the local causal pathway from \(x_{i}\) to \(t\), as shown in Figure 3\(b\)._  
Figure 2: Framework of the heuristic causal inference model. According to Definitions 3 and 4, if \(x_{i}\) is an end node in a global causal network (\(x_{i}\) has no parent), then the global causal pathway of \(x_{i}\) is the same as its local causal pathway. In Figure 3a, there are 9 global causal pathways from \(x_{3}\) to \(t\), including \(x_{1}\to x_{3}\to x_{5}\to t\), \(x_{1}\to x_{3}\to x_{6}\to t\), \(x_{1}\to x_{3}\to x_{6}\to x_{7}\to t\), \(x_{2}\to x_{3}\to x_{5}\to t\), \(x_{2}\to x_{3}\to x_{6}\to t\), \(x_{2}\to x_{3}\to x_{6}\to x_{7}\to t\), \(x_{3}\to x_{5}\to t\), \(x_{3}\to x_{6}\to t\) and \(x_{3}\to x_{6}\to x_{7}\to t\). In Figure 3b, there are 3 local causal pathways from \(x_{3}\) to \(t\), including \(x_{3}\to x_{5}\to t\), \(x_{3}\to x_{6}\to t\) and \(x_{3}\to x_{6}\to x_{7}\to t\). Thus, the global causal pathway of \(x_{3}\) contains its local causal pathway. **Definition 5**. : _Local Causal Pathway's Length--For a local causal pathway from \(x_{i}\) to \(t\), we define the total number of direct and indirect causes of \(t\) on this pathway as the length of the local causal pathway._  
In Figure 3b, the three local causal pathways from \(x_{3}\) to \(t\) have lengths of 2, 2, and 3, respectively. Based on Definitions 4 and 5, the average causal path length can be defined as follows:  
**Definition 6**. : _Average Local Causal Pathway's Length--We suppose there are \(y\) local causal pathways from \(x_{i}\) to \(t\), and for the \(wh\)\((w=1,2,\cdots,y)\) of them, its local causal pathway's length is \(d_{iw}\); and then, the average length of the local causal pathway from \(x_{i}\) to \(t\) is_  
\[d_{i}=\frac{\sum_{w=1}^{y}d_{iw}}{y}. \tag{1}\]  
In Figure 3, the average local causal pathway length from \(x_{i}\) to \(t\) is \((2+2+3)/3=7/3\). Based on Definitions 4 to 6, the causal pathway contribution degree of the cause variable to the target variable can be defined as follows:  
**Definition 7**.