# Probabilistic Models for Competence Assessment in Education
## 3 Results
### Probabilistic Models of Peer Assessment
#### 3.1.2 Tuned Models of Peer Assessment in MOOCs  
Looking for an answer to the same question posed by PAAS, the authors from _Tuned Models of Peer Assessment in MOOCs_[4] propose a Bayesian network model to represent the variables affecting the grade that a peer gives to another. Bayesian inference differentiates itself from other forms of inference by extending probability theory to the parameter space.  
In this case, the space over which the probability is allocated is \(Z\times\Theta\), where \(Z\) is the observational space and \(\Theta\) is the parameter space. Observations are denoted as \(z_{u}^{v}\in Z\), where \(u\) refers to the gradee and \(v\) to the grader. On the other hand, \(\Theta\) is the space parameter. More specifically, \(PG_{1}\)-bias considers that \(z_{u}^{v}\) is influenced by the following:  
1. The assignment's true score, \(s_{u}\in\mathbb{R}\). In the case of the implementation presented here, this is the teacher's grade.
2. The grader's bias, \(b_{v}\in\mathbb{R}\). This bias reflects a grader's tendency to either inflate or deflate their assessment by a certain number of percentage points. The lower these biases, the more accurate the grades will be.
3. The grader's reliability, \(\tau\in\mathbb{R}\), reflecting how close on average a grader's peer assessments tend to land near the corresponding assignment's true score after having corrected for bias. In this context, reliability is a synonym for precision or inverse variance of a normal distribution. Notice that the reliability of every grader is fixed to be the same value.  
The posterior probability distribution is computed as the product of the prior over the parameters times the likelihood function evaluated on the observations (times a constant depending only on the observations):  
\[\pi_{S}(s_{u},b_{v},\tau|z_{u}^{v})\propto\int_{\Theta}\pi_{S}(s_{u},b_{v},\tau )\pi_{S}(z_{u}^{v}|s_{u},b_{v},\tau)\mathrm{d}\theta \tag{8}\]  
The authors propose that the prior distribution \(\pi_{S}(s_{u},b_{v},\tau_{v})\) can be decoupled into three marginal probability distributions:  
\[\pi_{S}(s_{u},b_{v},\tau)=\pi_{S}(s_{u})\pi_{S}(b_{v})\pi_{S}(\tau) \tag{9}\]  
That is, the parameters are mutually independent. These priors have the form, according to [PERSON] et al. [4], of  
\[\pi_{S}(b_{v})=\mathcal{N}(0,\frac{1}{\eta_{0}}) \tag{10}\] \[\pi_{S}(s_{u})=\mathcal{N}(\mu_{0},\frac{1}{\gamma_{0}})\] (11) \[\pi_{S}(\tau)=\mathcal{G}^{-1}(\alpha_{0},\beta_{0}) \tag{12}\]  
Whereas the likelihood function is given by  
\[\pi_{S}(z_{u}^{v}|s_{u},b_{v},\tau)=\mathcal{N}(s_{u}+b_{v},\frac{1}{\tau}) \tag{13}\]  
The resulting PGM is shown in Figure 6.  
As Figure 6 shows, once the evidence \(z_{u}^{v}\) is injected, there is an active trail towards \(s_{v}\), which implies that we do not need an example of a teacher's grade to make inference about how they would grade any student in the class. This active trail does not mean, however, that we are not allowed to introduce some ground truths grades \(s_{v}\) in the Bayesian network. In that case, \(s_{v}\) is said to be a partially known parameter, and its injection should result in a reduction of the variance of the posterior distribution. Half-shaded nodes represent partially known parameters in \(BN\)s.