# Fault Inference of Electronic Equipment Based on Multi-State Fuzzy Bayesian Network
## 3 Multi-State Fuzzy BN  
The traditional Bayesian formula is shown as  
\[P(B_{i}|A)=\frac{P(B_{i})P(A|B_{i})}{\sum\limits_{j=1}^{n}P(B_{j})P(A|B_{j})} \tag{4}\]  
Figure 2: Flow chart of fault inference of multiple test indicators.  
where, \(B_{i}\) is the cause for event \(A\) and \(P(A|B_{i})\) is also known as the condition probability of \(A\) when event \(B_{i}\) occurs. \(P(B_{i})(i=1,2,\ldots)\) indicates the probability of cause \(B_{i}\), which is called prior probability, while \(P(B_{i}|A)\,(i=1,2,\ldots)\) shows the new understanding on cause \(B_{i}\) after event \(A\) occurs, which is known as posteriori probability.  
As a graphic network for probability inference based on Bayesian formula, BN is proposed to solve uncertainty of complex problems and incomplete problems [22; 23]. It has great advantages in solving faults caused by uncertainty and correlation of complex electronic equipment system [24]. Inference based on BN is to speculate the occurrence probability of some events at network nodes according to the structure of BN and CPT [25; 26; 27]. BN is used for bidirectional fault inference. In other words, it can evaluate reliability of a system from top to bottom and find the fault probability at source of the system through posteriori probability from bottom to top, so as to guide test personnel to find and rapidly locate faults based on modules or elements with large fault probability. In inference process, accurate inference algorithms--such as bucket-elimination algorithm, junction tree algorithm, and approximate calculation and inference algorithms, like search method and Gibbs sampling algorithm--are generally used. The former suits for inference based on BN with simple structure and small scale, while the latter is suitable for inference based on BN with complex structure and large amount of calculation [28; 29].  
The bucket-elimination algorithm solves the problem of joint probability distribution based on the idea of combinatorial optimization. The solving idea is to decompose joint probability into the product of CPTs through chain product for calculating joint probability and conditional independent characteristics [30; 31]. Binary nodes determined by fault probability include joint probability distribution \(P(S_{1},\,S_{2},\,\ldots,\,S_{n})\) of n nodes \((S_{1},\,S_{2},\,\ldots,\,S_{n})\).  
\[P(S_{1},\,S_{2},\,\ldots,\,S_{n})=\prod_{i=1}^{n}P(S_{i}|S_{1},\,S_{2},\, \ldots,\,S_{i-1}\,) \tag{5}\]  
After considering the conditional independence, that is, only taking into account of father node \(pa(S_{i})\) of node \(S_{i}\), the joint probability distribution can be simplified as  
\[P(S_{1},\,S_{2},\,\ldots,\,S_{n})=\prod_{i=1}^{n}P(S_{i}|pa(S_{i})\,) \tag{6}\]  
By performing factorization \(P(S_{i}|pa(S_{i})\,)\) on joint probability distribution, the node \(S_{j}\) without father node can be directly expressed as \(P(S_{j})\). According to elimination order, the nodes in the network are successively placed into a bucket and then the newly generated function factors are put into the next bucket to combine with the original variables. Such a process continues until the last bucket. Based on this, the obtained function factor is the calculated joint probability distribution.