This is the q(xi) distribution of information theory
(1/2) is bit information and $l_{i}$ is the length of the number of bit
then the cross enthropy derived to
I'm lack of experiecne to handling the log so can you show me how $-E_{p}[ln(q(x))/ln(2)]$ derived?
I want to know how
to
Change of base formula in logarithm states, $$\log_ba \times \log_xb=\log_xa$$
Then $$-\mathbb{E}_P\frac{\ln q(X)}{\ln2}=-\mathbb{E}\log_2 q(X)$$
rest is writing out the expectation explicitly.
The context is unclear, however as $q(x_i)=\frac{1}{2^{l_i}}$. Taking $\ln$ we obtain, $$\ln q(x_i)=l_i \ln 2^{-1} \implies l_i=-\frac{\ln q(x_i)}{\ln2}$$. Taking expectation on both sides with respect to the disribution $P$ it follows.