# User Mobility Modeling in Crowdsourcing Application to Prevent Inference Attacks
## 4 Proposed Approach
### POI Extraction
#### 4.5.1 Overview of Bayesian Networks  
A Bayesian Network (BN) consists of directed acyclic graphs (DAGs), where directed cycles are not allowed. The random variables are represented as vertices, and edges between nodes capture the dependence or causal relations. In addition to these, BNs can include special types of nodes known as decision nodes and utility nodes, which are primarily used in influence diagrams--an extension of BNs for decision-making processes. Decision nodes represent choices available to a decision maker, while utility nodes quantify the desirability of outcomes, allowing for evaluation based on different decision scenarios. Let \(V=\{X_{1},X_{2},\ldots,X_{N}\}\) be a set of variables represented as nodes in the network, which may include both chance nodes (random variables) and decision nodes (\(D_{1},D_{2},\ldots,D_{M}\)). The edges between these nodes denote directional influence, making node \(X_{i}\) a _parent_ of \(X_{j}\) if there is a direct edge from \(X_{i}\) to \(X_{j}\). Utility nodes (\(U_{1},U_{2},\ldots,U_{K}\)) do not directly influence other nodes but are influenced by combinations of chance and decision nodes to represent the utility associated with different outcomes. The absence of directed cycles ensures that the graph remains acyclic. The acyclic structure plays a crucial role in simplifying the joint probability distribution of the variables involved. Each node \(X_{i}\) within such a network is conditionally independent of any of its non-descendants when the values of its parent nodes \(\text{pa}(X_{i})\) are given. This conditional independence is foundational for defining the probability of observing any particular state of \(X_{i}\), which is calculated as follows:  
\[P(X_{i}\mid\text{pa}(X_{i}))\]  
where \(\text{pa}(X_{i})\) denotes the parents of \(X_{i}\). Building on this principle, the joint probability distribution for all the variables in the network is the product of these individual conditional probabilities, which is formally expressed as follows:  
\[P(X_{1},X_{2},\ldots,X_{N})=\prod_{i=1}^{N}P(X_{i}\mid\text{pa}(X_{i}))\]  
BNs are adept at handling a variety of variable types, including discrete, continuous, or a mixture of both, which allows them to be applied in diverse fields. Discrete variables in these networks are defined with a finite number of states, and their relationships are quantified using Conditional Probability Tables (CPTs), which detail the probabilities of one state occurring given the states of parent variables. Continuous variables, on the other hand, require a parametric form or a piecewise representation to define their conditional distributions accurately, reflecting the complexities involved in modeling real-world processes that change over a continuum. A primary application of BNs is belief propagation, which is a powerful method for updating the marginal probabilities of the variables based on new observations or evidence. This process calculates the marginal probability of a variable \(X_{i}\), given evidence \(e\), using the following formula:  
\[P(X_{i}\mid e)=\frac{P(X_{i},e)}{P(e)}\]  
where \(P(e)\) is the probability of the evidence. Belief propagation leverages the network's modular structure and the conditional independencies it represents to efficiently compute these probabilities, facilitating rapid updates to beliefs as new data become available.