9.5 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.

• It is a Bayesian approach that estimates an examinee’s proficiency across attributes by employing both observed item responses and prior information.

• EAP often provides more stable estimation than other methods (e.g., MLE), especially when sample sizes are small or the response data is sparse.

Specifically,

\[E(\alpha_{ik})=\sum_{c=1}^C\alpha_{ck}P(\mathbf{\alpha}_c|\mathbf{Y}_i),\] where, \(E(\alpha_{ik})\) represents the expected value of the latent attribute \(\alpha_{ik}\) for individual \(i\), with respect to the \(k-th\) attribute. It is the EAP estimate for the k-th attribute for individual \(i\). It gives the probability that examinee i has mastered attribute \(k\) considering both prior information and their observed responses.

If you are estimating whether a student has mastered “geometry” (attribute \(k\) based on their test responses, this equation computes the expected probability of mastery across all possible combinations of skills the student might have, weighted by the likelihood of each combination being true for that student.

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 \[\alpha_{ik}=I\big[E(\alpha_{ik})>0.5\big]\]