## 2 System Detectability Based on Bayesian Networks  
Optimal sensor deployment is \"mission-specific,\" meaning that the sensor-deployment strategy is closely related to specific objectives, such as decision-making and knowledge discovery [25]. This article focuses on the crucial issue of parameter estimation precision in wet-grinding systems. Wet grinding performance evaluation relies on estimating specific variables, such as model parameters or performance indicators. These particular variables can be determined via data reconciliation and monitoring; that is, adequate sensors must be allocated at the right places to guarantee the precision of the variables involved in the estimation scheme. Having all physical parameters sensed in a wet-grinding system is unnecessary and impossible because of their intrinsic dependence/correlation relationships. An effective method to represent the causality between physical parameters in a system for wet grinding performance evaluation is to create probabilistic networks or probabilistic graphical models, which are increasingly being viewed as a convenient high-level language for structuring a complex system of equations and an explicit representation of dependence/correlation between variables that ignores the specific functional details. BNs are a popular probabilistic network--a powerful method for presenting and reasoning with uncertainty. A valuable technique for BNs is to assume that the arcs represent causality. A BN is a directed acyclic graph that facilitates the representation of the causal relationships among physical variables in the wet-grinding system. A BN has two components: its structure and parameters. The structure, or topology, of a BN should capture qualitative relationships between physical variables. The design of a BN can be obtained from learned data or engineering knowledge. The former uses a series of statistical significance tests of conditional independence for structure learning [26]. The latter primarily depends on domain knowledge from human experts. In light of the practical experience of professional technicians in the field of wet grinding, the latter was adopted in this study to build the topology of the BN. In particular, two nodes should be connected directly with the arc, indicating the direction if one affects or causes the other. The nodes in a BN represent a set of physical variables, \(\mathcal{F}=\{F_{1},\,F_{2},\,\ldots,\,F_{n}\}\), from the wet-grinding system. A set of directed arcs connects the pairs of nodes, \(F_{i}\to F_{j}\), representing the direct dependencies between variables. If there is a directed arc from \(F_{i}\) to \(F_{j}\), then \(F_{i}\) is a direct cause (called a parent) of \(F_{j}\), denoted by \(\Re_{k}\big{(}F_{j}\big{)},\ k\in Z^{+}\). Otherwise, if there is an indirect path from \(F_{i}\) to \(F_{j}\), that is, \(F_{i}\rightarrow\cdots\to F_{j}\), then \(F_{i}\) is an indirect cause (called an ancestor) of \(F_{j}\), denoted by \(\Re_{k}\big{(}F_{j}\big{)},\ k\in Z^{+}\). The parameters of a BN are a set of conditional probability distributions to quantify the relationships between connected nodes. When the wet-grinding system runs under normal conditions, all physical variables are assumed to adhere to the standard normal distribution. Here, a linear Gaussian parameterization of the BN is employed to represent the relationship between the various physical variables [27], that is:  
\[F_{j}=\sum_{k=1}^{m_{j}}\delta\big{(}\Re_{k}(F_{j}),F_{j}\big{)}\cdot\Re_{k} \big{(}F_{j}\big{)}+\theta_{j}, \tag{1}\]  
where \(\delta(\cdot,\cdot)\) is a path coefficient. All the physical variables follow the standard normal distribution; thus, \(\theta_{j}\) and \(\Re_{k}\big{(}F_{j}\big{)}\) are independent of each other, \(\theta_{j}\)\(\sim\) N\(\left(0,\sigma_{j}^{2}\right)\); here, \(\sigma_{j}^{2}\)\(=1-\sum_{k=1}^{m_{j}}\delta\big{(}\Re_{k}(F_{j}),F_{j}\big{)}\cdot\rho\big{(} \Re_{k}\big{(}F_{j}\big{)},F_{j}\big{)}\), and \(\rho(\cdot,\cdot)\) is the Pearson correlation.