9.2 Maximum a Posterior (MAP) Estimation

Recall that a posterior distribution can be obtained from likelihood and prior distribution. Specifically, $$P\left({\alpha }_c|Y_i\right)=\frac{L({\alpha }_c;Y_i)p\left({\alpha }_c\right)}{{\sum }_{c=1}^{C}L({\alpha }_c;Y_i)p\left({\alpha }_c\right)}$$ The prior distribution, or $p\left({\alpha }_c\right)$, is supposed to be determined ‘’a priori’’, but this is usually challenging. We can use the estimated proportions $\hat{p}\left({\alpha }_c|X\right)$ instead. The MAP estimation for student $i$ is ${\alpha }_c$ that maximize $P\left({\alpha }_c|Y_i\right)$ or its logrithm.