# Probabilistic Models for Competence Assessment in Education
## 3 Results
### Probabilistic Models of Peer Assessment
#### 3.1.3 PG-Bivariate: A Bayesian Model of Grading Similarity  
We contribute in this section with a novel Bayesian model of peer assessment, \(PG\)-bivariate. The approach adopted here tries to reconcile the benefits of Bayesian inference with the concept of trust posed in PAAS [8]. The key idea is to use a probability distribution  
Figure 6: \(PG_{1}\)-bias [4]. to model the similarities as graders between any pair of peers in the system (teacher included). The chosen form for the probability density function modeling the similarities is a bivariate normal distribution. Figure 7 shows the graph representing the joint probability distribution of our model in a toy example with three students and a teacher, \(e\). The small, dark squares represent the relationship variables \(\mathcal{R}\) between graders. More specifically, consider a class with \(S\) students and a teacher \(\varepsilon\). Each student \(v_{i}\) assesses a number \(m_{i}\) of peer students. The allocation of assignments to graders is such that every student is assessed by the same number \(n\) of peers. Denoting the assessments by peer \(v\) as \(z^{\varepsilon}\), we wish to compute the grades that the teacher would give to each student, \(z^{\varepsilon}\). The model proposes that \(z^{\varepsilon}\) is sampled from a normal probability density function with parameters \(\mu_{v}\) and \(\sigma_{v}\). We could add complexity to the model and encode causal reasoning over these parameters by populating the layers above \(\mu_{v}\) and \(\sigma_{v}\). We limit, however, to this basic level of characterization, without adding any more variables influencing location or scale. Please notice that in this case, \(z\) are vector variables containing the assessments of each peer to her assigned revieeses. In the case of \(z^{\varepsilon}\), it is a partially known vector: making it a completely observed variable would imply that the teacher has assessed all the students in the class, which is not the case. In line with the concept of direct trust coined by PAAS' authors, we define a bivariate vector formed by the set of common assessments \(\{(z_{i}^{v_{1}},z_{i}^{v_{2}}),(z_{j}^{v_{1}},z_{j}^{v_{2}}),\ldots,(z_{k}^{ v_{1}},z_{k}^{v_{2}})\}\) between two peers \(v_{1}\) and \(v_{2}\), hereafter denoted as \(\mathcal{R}_{v_{1},v_{2}}\). This sequence is sampled from a bivariate normal distribution such that  
* The location vector, \(\vec{\mu}_{v_{1}v_{2}}=(\mu_{v1},\mu_{v2})\) is composed of each of the peers' location parameters separately. * The covariance matrix \(\Sigma_{v_{1}v_{2}}\) contains the individual variances in its diagonal. The off-diagonal components codify the correlations between \(v_{1}\) and \(v_{2}\) when grading. Putting it all together, the equations for \(PG\)-bivariate model are \[z^{v_{i}}\sim\mathcal{N}(\mu_{v_{i}},\sigma_{v_{i}})\quad\forall v _{i}\] (14) \[\mathcal{R}_{v_{i},v_{j}}\sim\mathcal{N}(\vec{\mu}_{v_{1}v_{j}}, \Sigma_{v_{i}v_{j}})\quad\forall v_{i},v_{j}\] (15) where \(\vec{\mu}_{v_{1}v_{j}}=(\mu_{v_{i}},\mu_{v_{j}})\), \(\Sigma_{v_{1}v_{j}}= \begin{pmatrix}\sigma_{v_{1}}^{2}&\sigma_{v_{1}v_{j}}\\ \sigma_{v_{1}v_{j}}&\sigma_{v_{j}}^{2}\end{pmatrix} \), and \((v_{i},v_{j})\) refers to a pair of graders having a set of evaluations in common. We propose the following prior distributions for the hyperparameters:  
\[\sigma_{v_{i}}\sim\text{Cauchy}(x_{0},\gamma) \tag{16}\] \[\mu_{v_{i}}\sim\mathcal{N}(\vec{z},\Sigma) \tag{17}\]  
Figure 7: \(PG\)-bivariate. As for the modeling of the covariance matrix \(\Sigma\), denoting by \(D\) a diagonal matrix whose elements are the square root of the elements in the diagonal of \(\Sigma\), \(D=\sqrt{Diag(\Sigma)}\), then the correlation matrix \(\Omega\) is related to the covariance matrix \(\Sigma\) by  
\[\Omega=D^{-1}\Sigma D^{-1} \tag{18}\]  
[PERSON] offers correlation matrix distributions, which have support on the Cholesky factors of correlation matrices.