9.3 Expected a Posterior (EAP) Estimation

Instead of finding \(\mathbf{\alpha}_c\) that maximizes \(P(\mathbf{\alpha}_c|\mathbf{Y}_i)\) or posterior distribution, we can also use the expected value as the estimate, which is referred to as the Expected a Posterior (EAP) estimation. Specifically, \[ E(\alpha_{ik})=\sum_{c=1}^C\alpha_{ck}P(\mathbf{\alpha}_c|\mathbf{Y}_i) \] Note that \(E(\alpha_{ik})\) is usually called mastery probability or the probability of mastering attribute \(k\) for student \(i\). It is a number between 0 and 1, but \(\alpha_{ik}\) must be either 0 or 1. We can define \[ \alpha_{ik}=I\big[E(\alpha_{ik})>0.5\big] \]