Nemer R.S.

SHEI “National Mining University”, Ukraine

Constructing a Bayesian network for students modeling

A Bayesian network is an acyclic oriented graph in which each vertex (node of the network) represents an n-valued variable, arcs denote the existence of immediate cause-effect relationships between the connected variables, and the strength of these dependencies is quantitatively expressed as the conditional probabilities associated with each of the variables.

Bayesian networks are one of the types of probabilistic graphical models.

To describe a Bayesian network, it is necessary to determine the structure of the graph and the parameters of each node. This information can be obtained directly from data or from expert assessments. This procedure is called learning of Bayesian network.

A Bayesian network is a common researchers’ choice to describe the fuzzy connection between student's achievements and their competences in many research projects. Models based on Bayesian networks have been used extensively in the development of computer-based learning tools since the end of the 1990s, especially by foreign researchers.

The structure of a Bayesian network reflects the structure of students' knowledge, and is an instrument through which it is possible to make judgments and assessments about the level of students’ preparedness, as well as to make decisions.

The attractiveness of Bayesian models lies in their high productivity, and also in an intuitive representation in the form of a graph.

The structure of the training course involves the division of discipline into chapters, and each of the chapters, in turn, corresponds to a set of concepts. Testing of students includes a set of test tasks, each of which can require the mastery of one or more concepts. In turn, the mastery of each of the concepts may be necessary to perform one or more test tasks.

Measuring the students’ competence level with their answers to test tasks is a typical problem of probabilistic reasoning. The two most common cases, in which there is uncertainty, are called slip and guess. Students may accidentally respond incorrectly to a question, the answer to which they know - this situation is called a slip. Also, students may accidentally guess the correct answer or write off the assignment. Such a case is called a guess.

The main stages of building a Bayesian network for students modeling are identification of variables, definition of the structure and definition of parameters.

Modeling should begin with the identification of variables that relate to the modeled domain. Variables can be divided into four classes according to their role in the model: target, evidences, factors, auxiliary.

1. target variables are used to model what is of interest. As a rule, target variables reflect latent characteristics. This means that there is no way to measure them directly. An example of a target variable in education is the student's understanding of a concept. This can not be measured directly, but only with the help of, for example, a test or an exam. The level of competence formation is the target variable.

2. variables of evidence are otherwise called observation variables. They are used to provide information on the target variables. In the modeling of students, evidences can be user actions. And it can be completely different levels of action - from clicking on a mouse button to performing a competency-oriented task.

3. factors are variables that model sources of influence on the target variable. They are also called context variables. Factors are divided into four categories according to their influence on the variable: promoters, inhibitors, requirements, exceptions. Promoters have a positive correlation with the target variable and help ensure that the characteristic is manifested. The inhibitors act on the contrary and have a negative correlation. Requirements are mandatory in order for the associated characteristic to manifest itself. The exceptions reduce the probability of the associated characteristic to zero.

4. auxiliary variables are used for convenience. For example, if a node has a lot of parent nodes, intermediate auxiliary variables can be used to group them. Due to this, the network structure is simplified, and the number of parameters is reduced.

After defining the variables, the next step in constructing a model is to define the structure. The network structure is determined by the arrangement of the edges between the variable nodes. As mentioned above, in the Bayesian network, the edges are directed. Changing the direction of the edge matters. The sense of an edge is that the variable in the initial vertex has a direct effect on the variable in the target. Thus, random events are connected by causal relations. In this regard, Bayesian networks are sometimes called causal. However, from a mathematical point of view, Bayesian networks do not necessarily speak of a causal relationship between variables. Often speak also about diagnostic communication between vertices in a network. The structure of the Bayesian network can be obtained directly from data or from expert assessments.

The final step in building a model is defining parameters. To do this, we need to specify a priori distributions for nodes that do not have parents (root nodes), and conditional probability distributions for all other nodes of the Bayesian network. As with the structure definition, the parameters can be specified by the expert, or obtained from the data. It is also possible to combine both of these approaches.

After the Bayesian network is constructed, it is ready to be able to perform calculations with its help.