Exercise 2.15 - MLE minimizes KL divergence to the empirical distribution

Expand the KL divergence:

We use the weak law of large numbers in the third step and drop the entropy of empirical distribution, which is independent of $𝜃$, in the last step. The other direction of optimization is $\arg <!--\; FUNCTION\; APPLICATION\; -->\underset{𝜃}{\min <!--\; FUNCTION\; APPLICATION\; -->}\left\{\mathit{𝕂𝕃}(q(𝜃)||p_{\text{emp}})\right\}$. It contains an expectation term w.r.t. $q(𝜃)$ and is harder to solve.