In the definition of jointly typical sets (in "Elements of Information Theory", ch. 7.6, p. 195), we use
$$-\frac{1}{n} \log{p(x^n)}$$ as the empirical entropy of an $n$-sequence with $p(x^n) = \prod_{i=1}^{n}{p(x_i)}$. I never came across this terminology before. It is not defined explicitly anywhere according to the index of the book.
My question basically is: Why is empirical entropy not $-\sum_{x}{\hat p (x) \log(\hat p(x))}$ where $\hat p(x)$ is the empirical distribution?
What are the most interesting differences and similarities between these two formulas? (in terms of properties they share/do not share).