# Probabilistic Models for Competence Assessment in Education
## 3 Results
### Probabilistic Models of Peer Assessment
#### 3.1.1 Personalized Automated Assessments  
Provided a community (a class) and a leader or special member from that community (the teacher), the aim of [PERSON] et al. [8] is to predict as accurately as possible the personalized assessments that a teacher in a class would make. The key idea behind this model is to unequally weigh the opinion of the members in the community (the students) so that the closer a student's marking style is to the teacher's style, the more relevant that student's opinion will be to predict the teacher's opinion. The closeness in marking styles will be represented as a trust value. That is, the teacher's trust in a student depends on the similarity between the teacher's (past) assess ments and the student's (past) assessments of the same assignments. In what follows, a one-to-one mapping between assessed students (_a.k.a._ grades) and assignments is presumed. In other words, for all the experiments in this work, every student completed one and only one assignment (_a.k.a._ exam). In the case there were no commonly assessed exams by the teacher and a given student, the authors propose a key concept: indirect trust. In short, an indirect trust measure is computed as the reputation of the student within the community, where this reputation is biased towards the teacher's perspective. Model  
Let \(\varepsilon\) represent a person who needs to assess a set \(\mathcal{I}\) of objects and let \(\mathcal{P}\) be a set of peers able to assess objects in \(\mathcal{I}\). In the context of our analysis, this would be a teacher that wants to assess a number of assignments (\(\mathcal{I}\)) completed by students that are evaluated by other students (\(\mathcal{P}\)). Assessments made by peers \(v\in\{\varepsilon\}\cup\mathcal{P}\) on an object \(u\in\mathcal{I}\), noted \(z_{u}^{v}\), are elements from an ordered evaluation space \(\mathcal{E}\). An automated assessment of \(\varepsilon\)'s opinion on an object \(u\), noted \(\varepsilon_{u}^{e}\), is a probability distribution \(\mathbb{P}=\{x_{1}\mapsto a_{1},x_{2}\mapsto a_{2},...,x_{n}\mapsto a_{n}\}\), where \(x_{i}\in\mathcal{E}\) and \(\alpha_{i}\in[0,1]\), with \(\sum_{i}\alpha_{i}=1\). A value \(\alpha_{i}\) represents the probability that \(\varepsilon\)'s true assessment of \(u\) is \(x_{i}\). Hence, the more peaked the grading distribution on an object is, the more confident we will be that the automated assessment closely approaches \(\varepsilon\)'s. Inversely, the flat, equiprobable distribution will be the one denoting ignorance of \(\varepsilon\)'s true assessment. Given a history of past peer assessments over \(u\), \(\mathcal{O}^{u}\), the ultimate goal of PAAS is to compute  
\[\mathbb{P}(X_{u}^{\varepsilon}=x|\mathcal{O}^{u})\]  
That is, we aim to compute the probability distribution representing \(\varepsilon\)'s evaluation on every object given the assessments of peers on that object. To that end, every individual evaluation \(z_{u}^{v}\) from \(\mathcal{O}^{u}\) is taken into account, and the probability \(\mathbb{P}(X_{u}^{\varepsilon}=x|z_{u}^{v})\) is computed as a function of the trust (expected similarity between previous assessments) that \(\varepsilon\) has on \(v\). In our context, this trust is computed using the assignments graded in common by the teacher, \(i\), and a student, \(j\). In the case that there are no assignments in common, an indirect trust measure is obtained based on the notion of transitive trust: the trust that \(i\) has on student \(j\) can be computed from the trust \(i\) has on student \(k\) and the trust that student \(k\) has on student \(j\). As there are many possible paths connecting \(i\) to \(j\), appropriate aggregation functions have to be provided. When two agents \(i\) and \(j\) have commonly assessed one or more objects, a direct trust relationship between them \(\mathbb{T}_{i,j}\) can be computed. This direct trust is modeled as a probability distribution on the difference between the evaluations performed by \(i\) and \(j\).