**2. Theoretical approaches**

The purpose of this section is to provide the readers with knowledge required in the field of learner modeling. In this section, we address the definitions and terminologies of the chapter's key words.

#### **2.1. Model of the learner**

#### *2.1.1. Definition*

Learner modeling is the modeling of all the important features that affect the learner (knowl‐ edge, preferences, goals, etc.). It identifies relevant information, and structures, initializes, updates and exploits it. By replacing the word "learning" with the term "user", this definition is also applicable to the model of the user. An application other than the learner's educational model is called the user model.

The main goal of a learner model is to store learner information, such as the learner's level of knowledge or skill pertaining to a given topic, and his or her personal information, such as psychological characteristics and preferences.

Zaitseva [1] defines the learner model as a set of structured information about the learning process, in which the characteristics of the learner are considered to be the values of this structure. According to Beck [2], the learner model acts as the key to system adaptation by providing the necessary data to other modules.

The uncertainty of the information contained within the learner model and the intention behind its creation have been the focus of many studies. Thus, a learner model represents system beliefs about learners' beliefs, accumulated during the diagnostic process.

The learner model can be an integral part of adaptive hypermedia systems, as it can be shared with multiple systems. In this last case, we discuss user modeling servers [3]. This type of server is used in environments where more distributed adaptive systems access the server to query or update user information. CUMULATE is one of the most known and used systems for user modeling servers.

#### *2.1.2. Foundations of the learner model*

learner with multiple objectives, from the evaluation of the learner's knowledge to the

Despite these various attempts at modeling learning characterized by a dynamic aspect, we always find that there are difficulties in achieving this goal. The proposed approaches provide us with only a static view of the learner model, yet this model is always in development (the learner's knowledge is evolving within the same module). Therefore, a dynamic view is essential. In order to monitor the behavior of the learner in real time and during formation,

The actions of the learner in a learning situation are not limited to valid or invalid actions (true and false), yet it is the actions that characterize the formation of the learning path. From this observation, we cannot represent information from the system of each learner using relative data. Rather, we must place our work in a probabilistic context due to changes in the learner

The problems presented in this chapter can be summarized as follows: How should we represent the different functions of a learner model? And what approaches can be used to

In this work, we propose the use of Bayesian networks as a probabilistic formalism to resolve the issue of management and dynamic update of the learner model. To resolve this issue, we must first ask: Why and how can we represent a learner model with Bayesian networks? How can we go from a dynamic representation of the Unified modeling language diagram of the model to a probabilistic representation with Bayesian networks? Is this consideration experi‐

The purpose of this section is to provide the readers with knowledge required in the field of learner modeling. In this section, we address the definitions and terminologies of the chapter's

Learner modeling is the modeling of all the important features that affect the learner (knowl‐ edge, preferences, goals, etc.). It identifies relevant information, and structures, initializes, updates and exploits it. By replacing the word "learning" with the term "user", this definition is also applicable to the model of the user. An application other than the learner's educational

The main goal of a learner model is to store learner information, such as the learner's level of knowledge or skill pertaining to a given topic, and his or her personal information, such as

we must adopt a dynamic modeling approach when managing learner modeling.

perform updates on the different characteristics of such a model?

recognition of the plan followed in problem solving.

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

model during formation.

mentally justified?

key words.

*2.1.1. Definition*

**2. Theoretical approaches**

**2.1. Model of the learner**

model is called the user model.

psychological characteristics and preferences.

Self [4] defined a formalization of the learner model that is based on the beliefs and knowledge of the system and the learner. Beliefs are represented by formulas in propositional calculus. Objects of belief are called propositions. Beliefs are related to the behavior of an agent (A), a user (U) or a system (S). BA = {p/BAP} is the set of beliefs of agent A. BAP are the proposals themselves.

BSU = {p / BSBUp} is the set of proposals that system S believes are believed by user U (see Fig. 1).

**Figure 1.** The representation of the system and user beliefs

The learner model can be defined by a set of proposals that the system S thinks about learner U: UM = BS (U) = {p / BSp (U)}

Belief can be replaced by knowledge; therefore, KAP = BAp.

To distinguish between the different aspects of the learner model, Self distinguishes the following proposals:


This proposal describes the cognitive and personal characteristics of the learner, also known as behavioral skills, which include preferences, tasks, goals and experience.

#### **2.2. Bayesian networks**

Before describing our investigation of the use of Bayesian networks in learner modeling, we'll define such networks and address the meaning of inference in this context.

In the rest of this section, we'll take a typology of nodes inspired by Conati [5], and found in different terms in the literature. The field layer is the set of nodes modeling epistemic knowl‐ edge of the learner, and the task layer is the set of nodes modeling the actions of the learner.

#### *2.2.1. Definition*

Numerous models have been created through the representation of knowledge. Probabilistic graphical models, and especially Bayesian networks initiated by Pearl [6] in the 1980s, have proven to be useful tools for representing uncertain knowledge and reasoning from incomplete information.

A Bayesian network is a directed acyclic graph in which the nodes correspond to the variables (user properties), and the links represent probabilistic relationships of influence. These variables can belong to the field of knowledge, the base knowledge and / or the cognitive model. Each node represents the system's belief about possible values (levels, states) of the variable. Thus, the conditional probability distribution must be specified for each node. If the variables are discrete, they can be presented as a table.

The graph is also called the "structure" of the model, and the probability tables are its "param‐ eters". They can be provided by experts, or calculated from data; generally speaking, the structure is defined by experts and the calculated parameters are from experimental data.

#### Consider a Bayesian network *B* =(*G*, *N* ) defined by

*G* =(*X* , *E*), an acyclic directed graph with various vertices associated with a set of random variables *X* =(*X* , ..., *Xn*) ;N= {P(Xi | Pa(Xi))} All the probabilities of each node *Xi* are con‐ ditional to the state of its parents Pa(Xi) in G.

According to Mayo [7], a Bayesian network allows compact representation of the joint probability distribution over a set of variables:

$$\mathbf{P(X1,X2,\cdots,Xn)} = \prod\_{i=1}^{n} \mathbf{P(Xi \mid \mid \mid \mathbf{Pa(Xi)})},$$

These methods obviously use the concept of conditional probability, i.e., what is the probability of *Xi* knowing that I have observed *Xj* ; but they also use the Bayes theorem, which calculates, conversely, the probability of *Xj* knowing *Xi*, when P(Xi | Xj) is known.

#### *2.2.2. Bayesian network construction*

To specify a Bayesian network in a comprehensive way, it is necessary, as we have seen in the definition, to specify the network structure (the acyclic graph) and the network parameters (the probability tables). To reach this specification, there are two approaches: 1) the collection of expertise, and 2) the machine learning, which is one of the attractions of Bayesian networks. A combination of these two approaches is also possible.

In the first approach, the collection of expertise, we must begin by defining the network structure, starting with identifying the possible nodes, and then we distinguish between hypothetical (unobservable) variables and informational (observable) variables. The next step concerns the analysis of the existing arc in terms of the influence of one variable upon another. Traditionally, if an arc is directed from A to B, A is a cause of B; however, in the case of learner modeling, we will see that the interpretation is not so simple. The parameters are in turn attached to approximations using qualitative or frequentists' information.

A Bayesian network is considered as a probability distribution. By using maximum likelihood as a statistical learning parameter criterion, the result is a Bayesian network with a fixed structure and with E as a comprehensive basis of example. If the parameters of the Bayesian network are equal to the frequencies of the same features observed in E, the maximum likelihood will be achieved. A test is necessary to determine the conditional independence of random variables in the statistical learning structure.
