9.4 Maximum a Posterior (MAP) Estimation

The Maximum-A-Posteriori (MAP) estimation procedure for attribute estimation in CDM is a Bayesian approach that estimates an examinee’s attribute profile by identifying the profile with the highest posterior probability, given the observed item responses and prior information.

MAP focuses on finding the most likely (or “maximum”) profile rather than averaging over all possible profiles.

Recall that a posterior distribution can be obtained from likelihood and prior distribution. Specifically,

\[ P(\mathbf{\alpha}_c|\mathbf{Y}_i)=\frac{L(\mathbf{\alpha}_c;\mathbf{Y}_i)p(\mathbf{\alpha}_c)}{\sum_{c=1}^CL(\mathbf{\alpha}_c;\mathbf{Y}_i)p(\mathbf{\alpha}_c)} \] The prior distribution, or \(p(\mathbf{\alpha}_c)\), is supposed to be determined ‘’a priori’’, but this is usually challenging. We can use the estimated proportions \(\hat{p}(\mathbf{\alpha}_c|\mathbf{X})\) instead.

The MAP estimation for student \(i\) is \(\mathbf{\alpha}_c\) that maximize \(P(\mathbf{\alpha}_c|\mathbf{Y}_i)\) or its logarithm.

• The MAP estimate is the attribute profile with the highest posterior probability. This is the most likely profile for the examinee, given the observed data and the prior information.

• Instead of averaging over all possible profiles (as in EAP), MAP selects the profile with the maximum posterior probability.

MAP is simple to apply, provide direct estimates for attributes, and often applicable for large data. However, MAP is sensitive to prior, and less stable for sparse data.