I have the following expression: $$\int\int p(z|x)\cdot log\left(p(z)\right)dzdx$$ In order to compare it to another expression that is also integrated over another RV y, I would like to marginalize it over y by using total probability. However, I'm not sure if it suffices to add the new conditioning on the non log term: $$\int\int\int p(z,y|x)\cdot log\left(p(z)\right)dzdxdy$$ Both probabalistic terms: $$\int\int\int p(z,y|x)\cdot log\left(p(z|y)\right)dzdxdy$$ Or do I need to use integration by parts?
Thanks! :)