# Extracting Searching as Learning Tasks Based on IBRT Approach
## 3 Data and Methods
### Modeling Progress
#### 3.3.1 Improved Bayesian Rose Tree Model  
The IBRT model is based on the BRT model used to construct multibranch hierarchies. [PERSON] et al. (2018) first proposed an efficient Bayesian nonparametric model for discovering hierarchical community structure in social networks. BRT is based on a greedy probabilistic agglomeration approach to construct multibranch hierarchies. [PERSON] et al. (2018) first proposed the BRT (Bayesian rose tree) model for extracting hierarchies of tasks and subtasks. Although the traditional BRT algorithm can achieve a hierarchical representation of search tasks, the BRT model cannot be used for SAL task representation as it cannot take learning outcomes and learning styles into account. To solve these problems, we propose the IBRT model for the representation hierarchies of SAL tasks. IBRT can sense both searching features and learning features, further providing accurate SAL task representation on the basis of learning styles. The authors of (2018, 2018) pointed out that the query is the starting point of a SAL process, and it can reflect learning objectives and learning structure. Furthermore, the authors of (2018) proposed that queries can reflect the learning tasks of the learner. Therefore, we use query sets to represent the SAL task-subtask structures. Specifically, we first propose Equation (1) in the IBRT model to build the representation of SAL tasks. \[p(D_{m}|T_{m})=\pi_{LKN}g(D_{m}|T_{m})+(1-\pi_{LKN})\left(\pi_{Tm}f(D_{m})+(1- \pi_{Tm})\prod_{T_{i}\in ch(T_{m})}p(D_{i}|T_{i})\right), \tag{1}\]where \(D_{m}=\{Q_{1},Q_{2},\ldots,Q_{m}\}\) is the set of issued queries. The queries in \(D_{m}\) are sorted by the time sequence, where \(Q_{i}\) is the \(i\)-th issued query. \(p(D_{m}|T_{m})\) is the probability of data \(D_{m}\) given a partitioning by the tree \(T_{m}\). \(g(D_{m}|T_{m})\) is the marginal probability for \(D_{m}\) which is partitioned by \(T_{m}\) when the user's learning style is LKN. \(\pi_{LKN}\) is the prior probability of the learners whose learning style is LKN, whereas \((1-\pi_{LKN})\) is the prior probability of the learners whose learning style is LIN. \(\pi_{Tm}\) is the mixture probability, which represents the prior probability that all data points in \(T_{m}\) belong to a single cluster. \(f(D_{m})\) is the marginal probability of the data \(D_{m}\). \(p(D_{i}|T_{i})\) is a hyperparameter representing the probability that the nodes in query set \(D_{i}\) are partitioned given the tree \(T_{i}\). At the beginning of the modeling process, each query in \(D_{m}\) is considered an independent tree. Specifically, \(T_{i}=\{x_{i}\}\), where \(x_{i}\) is the \(i\)-th query in \(D_{m}\). A query may independently be a task or part of a task. We recursively define the representation of SAL task hierarchy, where a tree \(T\) may contain queries \(T=\{x_{1},\ x_{2},\ldots,x_{n}\}\) or may be split into subtrees \(T=\{T_{1},T_{2},\ldots,T_{n}\}\), where each \(T_{i}\)\((1<i<n)\) is a subtree in \(T\). This allows us to consider \(T\) as a hierarchical tree structure for the task-subtask relationship, where tasks and subtask are represented by query sets. During the modeling process, for each step, the IBRT method chooses two trees \(T_{i}\) and \(T_{j}\) from \(T\) and merges them into a new tree \(T_{m}\). In this paper, we adopt the same tree merging strategy along with BRT, as shown in Figure 2. Inspired by the study proposed by [PERSON] and [PERSON] [13], at each step, IBRT greedily finds and merges two trees when the following formula is maximum:  
\[\frac{p(D_{m}|T_{m})}{p(D_{i}|T_{i})p(D_{j}|T_{j})}, \tag{2}\]  
where \(p(D_{m}|T_{m})\) is the possibility of the partition of query set \(D_{m}\) given the tree \(T_{m}\). All queries in \(D_{m}\) are leaves of tree \(T_{m}\), and \(p(D_{m}|T_{m})\) is calculated recursively through its child nodes. Calculating the marginal probability for LKN  
For learners whose learning style is LKN, \(g(D_{m}|T_{m})\) is calculated according to the knowledge network.