# Probabilistic Models for Competence Assessment in Education
### Representation  
A Bayesian network (BN) is a directed acyclic graph \(G=(V,E)\) and a set of parameters \(\Theta=\{\theta_{1},\theta_{2},...,\theta_{n}\}\) labeling the probability distributions associated with the nodes in the graph. This graph represents a decoupled joint probability distribution, such that  
Figure 1: Iterative addition of variables to a generative model. This methodology constructs a joint probability distribution from intermediate conditional probability distributions.  
Figure 2: Reverse construction of a joint probability distribution. In this case, we begin with a marginal distribution over some few variables and then increase the complexity by conditioning this distribution over a new set of (ancestor) nodes.  
The vertices represent the random variables that we model.  
For each vertex \(X_{i}\), there is a conditional probability distribution \(\pi(X_{i}|\mathbf{pa}_{i})\).  
The chain rule for Bayesian networks states that a Bayesian network \(M=(G,\Theta)\) can be expressed as the product of marginal and conditional probability distributions associated to its nodes:  
\[\pi_{M}(\overrightarrow{X})=\prod_{i=1}^{n}\pi(X_{i}|\mathbf{pa}_{i};\Theta_{ i}) \tag{1}\]  
where \(\overrightarrow{X}=\{X_{1},X_{2},...,X_{n}\}\) and \(\mathbf{pa}_{i}\) stands for _parent_ nodes and denotes the set of variables with a direct edge towards \(X_{i}\). The symbol \(\Theta_{i}\) represents the parameters of the probability distribution associated with that same vertex. If a node has no parents, then the probability distribution associated with it is marginal. As a side note, when depicting a Bayesian network, observed nodes will be shaded.  
In addition, some representations will introduce the plate notation, a method of representing variables that repeat in a graphical model. Instead of drawing each repeated variable individually, a plate (rectangle) is used to group variables into a subgraph that repeat together, and a quantity is drawn on the plate to represent the number of times the subgraph repeats in the plate. It is assumed that the subgraph is duplicated that many times, the variables in the subgraph are indexed by the repetition number, and any links that cross a plate boundary are replicated once for each subgraph repetition.  
Before deepening into the dependencies between variables in a Bayesian network, let us introduce some important concepts that will be used in the presentation of the models implemented in this research, namely parents, children, ancestors, ancestral ordering, and Markov blanket.  
Given a node \(X\) in a Bayesian network, its parent nodes are the set of nodes with a direct edge towards \(X\).  
Given a node \(X\) in a Bayesian network, its children nodes are the set of nodes with an incoming edge from \(X\).  
Given a node \(X\) in a Bayesian network, its ancestors are given by the set of all variables from which we can reach \(X\) through a directed, arbitrarily long path.  
Given the set of all the variables modeled in a Bayesian network, \(X=\{X_{1},X_{2},...,X_{N}\}\), an ancestral ordering of the variables is followed when traversing the network; if every time we reach a variable \(X\), we have already visited its ancestors.  
Given a node \(X\) in a Bayesian network, its Markov blanket is given by its parents, its children, and the parents of its children.