# Probabilistic Models for Competence Assessment in Education
## 3 Results
### Probabilistic Models of Peer Assessment
#### 3.1.1 Personalized Automated Assessments
Let \(\mathbb{P}(X_{u}=x)\) for \(x=\mathit{diff}(i,j)\) be the probability distribution of the assessment difference between \(i\) and \(j\). The new assessment must be reflected in a change in the probability distribution. In particular, \(\mathbb{P}(X_{u}=x)\) is increased a fraction of the probability of \(X\) not being equal to \(x\): \[\mathbb{P}(X_{u}=x)=\mathbb{P}(X_{u}=x)+\gamma\cdot(1-\mathbb{P}(X_{u}=x))\] (5)For instance, if the probability of \(x\) is 0.6 and \(\gamma\) is 0.1, then the new probability of \(x\) becomes \(0.6+0.1\times(1-0.6)=0.64\). As in the example, the value of \(\gamma\) must be closer to 0 than to 1, for considerable changes can only be the result of information learned from the accumulation of many assessments. 3. The resulting \(T_{i,j}\) is then normalized by computing the distribution that respects the new computed value and has a minimal relative entropy with the previous probability distributions: \[ \begin{array}{c}\mathbb{T}_{i,j}(X)=\text{argmin}_{\mathbb{P}^{\prime}} \mathbb{P}^{\prime}(X)\sum_{x^{\prime}}p(X^{\alpha}x^{\prime})\log\frac{p(X^{ \alpha}x^{\prime})}{p(X^{\alpha}x^{\prime})}\\ \text{such that }\quad\{p(X^{\alpha}x)=p^{\prime}(X^{\alpha}x)\}\end{array} \] (6) where \(p(X^{\alpha}x^{\prime})\) is a probability value in the original distribution, \(p^{\prime}(X^{\alpha}x^{\prime})\) is a probability value in the potential new distribution \(\mathbb{P}^{\prime}\), and \(\{p(X^{\alpha}x)=p^{\prime}(X^{\alpha}x)\}\) specifies the constraint that needs to be satisfied by the resulting distribution. These direct trust distributions between peers are stored in a matrix \(\mathcal{C}\). 4. To encode the decrease in the integrity of information with time (_information decays with time, I may be sure today about your high competence playing chess, but maybe in five years time I will be no longer sure if our interactions stop. You might have lost your abilities during that period_), the direct trust distributions in \(\mathcal{C}\) are decayed towards a decay limit distribution after a certain grace period. In our case, the limit distribution is the flat equiprobable \(\mathbb{F}\). When a new evaluation updates a direct trust distribution \(T_{i,j}\), \(T_{i,j}\) is first decayed before it is modified. 5. The indirect trust distributions between \(\epsilon\) and each peer are stored in a distributions vector \(\overrightarrow{t_{\epsilon}}\). Initially, \(\overrightarrow{t_{\epsilon}}\) contains the probability distributions describing ignorance \(\mathbb{F}\). When matrix \(\mathcal{C}\) is updated, \(\overrightarrow{t_{\epsilon}}\) is also updated as a product of its former version times matrix \(\mathcal{C}\): \[t_{\epsilon,j}^{k+1}=\bigoplus_{0<i\leq n}\mathbb{T}_{i,j}\otimes\mathbb{T}_{ \epsilon,i}^{k}\] (7)
6. If a direct trust distribution \(T_{\epsilon,j}\) exists between \(\epsilon\) and \(j\), the indirect trust distribution \(t_{\epsilon,j}\) is overwritten with \(T_{\epsilon,j}\) after the update of the indirect trust distributions. Please notice that the decay in the direct distributions \(T_{i,j}\) affects the indirect distributions \(t_{\epsilon,j}\) as well. This is because the indirect trust distributions are computed as aggregations of combinations of direct trusts.