9.3 Expected a Posterior (EAP) Estimation

Instead of finding ${\alpha }_c$ that maximizes $P\left({\alpha }_c|Y_i\right)$ 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, $$E\left({\alpha }_{ik}\right)=\sum _{c=1}^C{\alpha }_{ck}P\left({\alpha }_c|Y_i\right)$$ Note that $E\left({\alpha }_{ik}\right)$ 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 $${\alpha }_{ik}=I[E\left({\alpha }_{ik}\right)>0.5]$$