**4. Bayesian network development**

In Fig. 3, a main actor is identified, named "the learner". The figure shows the generalization relationships between use cases and the learner, and the generalization relationships of

In particular, the functional requirement of "Learner" represents all information about the learner in the hypermedia system (the learner's knowledge, skills, personal information, etc.). This functional requirement is shown with a generalization relationship with three functional

**•** "**Pretest**" – this represents information about the pretest the learner has to take before entering the learning situation. The pretest is composed of two types of evaluation compo‐ nents: 1) tests of knowledge depicted with the functional requirement "*knowledge*", and 2) the functional requirement "*skills*", which represents the test through which we will evaluate

**•** "**Learning Activity**" – this functional requirement represents information about the learning activities. Each learning activity in an adaptive educational hypermedia system is of two types: 1) static activities represented by the functional requirement "*Static*", and 2) interactive

**•** "**Evaluation**" – this represents the information about the evaluation tests the learner has to take after completion of each learning activity. If the learner fails in the evaluation, the learner must pass to remediation; which is represented by the functional requirement "*Remediation*", which is connected to the functional requirement "*Evaluation*" through an

In the case of remediation, the functional requirement "Remediation" involves activation of the functional requirement "Call Tutor" through an inclusion relation. This requirement

activities represented by the functional requirement "*Interactive*".

inclusion and extension between use cases.

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**Figure 3.** Use case diagram UML representing the learner actions

requirements:

the learner's skills.

extension of relationship.

In this section, we present the transformation of our use case diagram representing the learner model, as presented in [9], into a Bayesian network.

#### **4.1. The development rules of a Bayesian network**

#### *4.1.1. The generalization relationship transformation*

A generalized type of use case represents a functionality that allows all instances of specialized use cases. The transformation of this type of relationship to nodes of a Bayesian network is considered simple.

In Fig. 4, use case A is a generalization of use cases A1 and A2, and we represent the functional requirements of A1 and A2 as being descendant of the functional requirement A.

**Figure 4.** The generalization relationship transformed into a Bayesian network fragment

This results in a Bayesian network with a similar structure. The direction of the arc flows from A to A1 and A2 reflects top-down decomposition. This indicates that one is more likely to encounter a general case with specific functional requirements, including those in the Bayesian network, having developed the information that is represented by the arrows of the use case. Therefore,

$$P(A) = prior$$

$$P(A1 \mid A) = P(A \mid A1)P(A1) / \ P(A)$$

$$P(A2 \mid A) = P(A \mid A21)P(A2) / \ P(A)$$

#### *4.1.2. The inclusion relationship transformation*

The inclusion relation in a use case diagram represents the situation in which a use case is composed of a number of various use cases. For inclusion, a high level of use cases cannot be executed without the implementation of sub use cases.

Figure 5 represents this relationship, with case A including cases A1 and A2 if the behavior described by case A includes descendant behavior; that is, if A depends on A1. When A is pressed, the east must, as part of A.

**Figure 5.** The inclusion relationship transformed into a Bayesian network fragment

This results in a Bayesian network with a similar structure. The direction of the arc flowing from A to A1 and A2 represents a bottom-up composition. This indicates that one is more likely to encounter a general case with specific functional requirements, and the functional require‐ ments, including those in the Bayesian network, have developed the information that is represented by the arrows of the use case. Therefore,

$$P(A) = prior$$

$$P(A1 \mid A) = P(A \mid A1)P(A1) / P(A)$$

$$P(A2 \mid A) = P(A \mid A21)P(A2) / P(A)$$

#### *4.1.3. The extension relationship transformation*

This results in a Bayesian network with a similar structure. The direction of the arc flows from A to A1 and A2 reflects top-down decomposition. This indicates that one is more likely to encounter a general case with specific functional requirements, including those in the Bayesian network, having developed the information that is represented by the arrows of the use case.

( )

= =

*P P*

174 E-Learning - Instructional Design, Organizational Strategy and Management

executed without the implementation of sub use cases.

**Figure 5.** The inclusion relationship transformed into a Bayesian network fragment

*4.1.2. The inclusion relationship transformation*

pressed, the east must, as part of A.

(A1 | A) P(A |A1)P(A1) / P(A) (A2 | A) P(A |A21)P(A2) / P(A)

The inclusion relation in a use case diagram represents the situation in which a use case is composed of a number of various use cases. For inclusion, a high level of use cases cannot be

Figure 5 represents this relationship, with case A including cases A1 and A2 if the behavior described by case A includes descendant behavior; that is, if A depends on A1. When A is

This results in a Bayesian network with a similar structure. The direction of the arc flowing from A to A1 and A2 represents a bottom-up composition. This indicates that one is more likely to encounter a general case with specific functional requirements, and the functional require‐

*P A prior*

=

Therefore,

The extension relationship is probably the most useful because it has semantic meaning; it represents a particular use case branched additional behavior, given the satisfaction of certain conditions.

Figure 6 represents use case A, which extends to use case A1, when use case A can be called during execution in the case of A1 use. Run A1 can possibly lead to the execution of A; unlike the inclusion, the extension is optional.

**Figure 6.** The extension relationship transformed into a Bayesian network fragment

The additional criterion is described in the flow of events in textual description as another functional requirement node. The direction of the implication is an additional criterion (AC) to the functional requirement A1, and functional requirements are included in the Bayesian network, with information represented in cases of use arrows. Therefore:

$$\text{P(A1 \mid A, AC)} = \frac{\text{P(A \mid A1, AC)P(A1 \mid AC)}}{\text{P(A \mid AC)}}$$

#### **4.2. The Bayesian network developed**

The development of a Bayesian network based on the use case diagram for modeling the learner in an adaptive educational system passes through two essential steps:

#### *4.2.1. Specification of the model structure*

Taking the case of the node "Learner" to illustrate the stages of development of our Bayesian network representing the learner model, note that this node has three parent nodes (Pretest, Learning Activity and Evaluation), and that each of these nodes is composed of child nodes. Links to these nodes are prerequisite relationships:

	- **◦** Knowledge: the student must answer more than ten questions to measure his or her wealth of knowledge. This type of evaluation reflects the evaluated portion of knowledge of the learner.
	- **◦** Skills: This is written proof of whether the student can apply the knowledge gained in the module. This type of evaluation reflects the skills portion of the learner.

The value measuring the relative importance of each condition varies from 0 to 1, and the values of each evaluation element are defined by the teacher, who in this case is the teacher of the module "Database".

The relationship between the target variable (T) and the evidence variable (E) move from T to E, because the process that calculates the posterior probability of the target variable is the proof of knowledge of the diagnosis. Therefore, if the evidence variable has no children, the parents must be the target variables. There are two types of relationships:


#### *4.2.2. The specification of variable values*

Once the use case diagrams have been created, it is easy to create the structure of the Bayesian network using the rules described in previous sections. Figure 7 represents the Bayesian network constructed from the use case diagram shown in the previous section. Notice how conditional independence was directly modeled by applying the rules as shown.

In the Bayesian network developed, we observe that the node learner (L) has three parents: Learning Activity (A), Evaluation (E) and Pretest (T), which in turn correspond to three weights of prerequisite relationship: w1 = 0.1, w2 = 0.5, w3 = 0.4. Conditional probability of (L) is computed as follows:

$$P\left(L\vert A, E, T\right) = w\mathbf{1}^\*h\mathbf{1} + w\mathbf{2}^\*h\mathbf{2} + w\mathbf{3}^\*h\mathbf{3}$$

Where

**4.2. The Bayesian network developed**

*4.2.1. Specification of the model structure*

of the learner.

the module "Database".

Links to these nodes are prerequisite relationships:

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essential to guide the course of the learner.

**•** Prerequisite relations between target variables.

*4.2.2. The specification of variable values*

of two types, static and interactive, in the adaptive system.

course, The pre-test consists of two types of evaluations:

must be the target variables. There are two types of relationships:

The development of a Bayesian network based on the use case diagram for modeling the

Taking the case of the node "Learner" to illustrate the stages of development of our Bayesian network representing the learner model, note that this node has three parent nodes (Pretest, Learning Activity and Evaluation), and that each of these nodes is composed of child nodes.

**• Learning Activity –** In this node, all students following the course must go through activities

**• Pretest –** All learners must take a pre-test before engaging in the learning activities of each

**◦** Knowledge: the student must answer more than ten questions to measure his or her wealth of knowledge. This type of evaluation reflects the evaluated portion of knowledge

**◦** Skills: This is written proof of whether the student can apply the knowledge gained in

**• Evaluation** –After the student follows the learning activity, an evaluation is conducted to determine the student's level of knowledge and skill within the module. The evaluation is

The value measuring the relative importance of each condition varies from 0 to 1, and the values of each evaluation element are defined by the teacher, who in this case is the teacher of

The relationship between the target variable (T) and the evidence variable (E) move from T to E, because the process that calculates the posterior probability of the target variable is the proof of knowledge of the diagnosis. Therefore, if the evidence variable has no children, the parents

**•** Diagnostic relations of target variables to evidence variables. The control of concepts (targets) affects confidence of evidence. However, if the learner has failed a test, it is unclear if this is due to his lack of knowledge or ability, because there can be an unexpected error.

Once the use case diagrams have been created, it is easy to create the structure of the Bayesian network using the rules described in previous sections. Figure 7 represents the Bayesian network constructed from the use case diagram shown in the previous section. Notice how

conditional independence was directly modeled by applying the rules as shown.

the module. This type of evaluation reflects the skills portion of the learner.

learner in an adaptive educational system passes through two essential steps:

$$h\mathbf{1} = \begin{cases} \mathbf{1} \text{ if } A = L\\ \mathbf{0} \text{ otherwise} \end{cases}$$

$$h\mathbf{2} = \begin{cases} \mathbf{1} \text{ if } E = L\\ \mathbf{0} \text{ otherwise} \end{cases}$$

$$h\mathbf{3} = \begin{cases} \mathbf{1} \text{ if } T = L\\ \mathbf{0} \text{ otherwise} \end{cases}$$

We should state that {L, A, E, T} is a complete set of mutually exclusive variables, each of which is also a random and binary variable.

**Figure 7.** The Bayesian network developed of the learner model

Generalizing the formula below, we state that:

$$P\left(X = 1 \mid Y1, Y2...Yn\right) = \sum\_{i=1}^{n} w i^\* h i^\*$$

where *<sup>h</sup>* 1={ <sup>1</sup> *if Yi* <sup>=</sup> *<sup>X</sup>* <sup>0</sup> *otherwise* with given random binary variables X and Yi. Obviously, *P*(*not X* |*Y* 1, *Y* 2, ..., *Yn*)=1−*P*(*X* |*Y* 1, *Y* 2, ..., *Yn*).

**a.** The Conditional Probability Table of the Node "Learner"

Table 2 represents the CPT of each child node of the parent node Learner.


**Table 2.** The conditional probability table of "Learner" node

Because concepts A, E, and T have no prerequisite knowledge for understanding, their CPTs are specified as prior probabilities obeying uniform distribution, as stated in Table 3 (assigned medium value of 0.5 in most cases).


**Table 3.** The conditional probability table of "Learner" parents

**b.** The Conditional Probability Table of the Node "Pretest"

Table 4 represents the CPT of each child node of the parent node Pretest.


**Table 4.** The conditional probability table of "Pretest" node

**c.** The Conditional Probability Table of the Node "Learning Activity"


### Table 5 represents the CPT of each child node of the parent node Learning Activity.

**Table 5.** The conditional probability table of "Learning Activity" node
