9.2 Maximum a Posterior (MAP) Estimation

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 logrithm.