Entropy of X given Y given Z

Question

I want to compare $H(X|Y)$ and $H(X|Y,Z)$.
It's known that $H(X) \ge H(X|Y)$. Can I say that $H(X|Y) \ge H((X|Y)|Z) = H(X|Y,Z)$?

Answer 1

Given three random variables $X,Y,Z$ consider,

$$I(X:Z|Y)=H(X|Y)-H(X|Y,Z)$$

where $I$ is Kulback Leibler Divergence. By positivity of KL divergence, we have

$$H(X|Y)-H(X|Y,Z) \ge 0$$