I want to compare between $$ \log P(x)= \int P(y)\log P(x)dy, $$ and $$ \mathbb{E}_y[\log P(x|y)] := \int P(y)\log P(x|y)dy. $$
I consider the setting that $X$ is a discrete, and $Y$ is continuous. How can I compare the size between this two quantities? Any clues would be very appreciated.