Basic Entropy Inequality and Identity question

Question

This is a solution to a problem I am working on: \begin{equation} \begin{aligned} H(X|Y) + H(Y|Z) &\ge^? H(X|Y, Z) + H(Y|Z) \\ &=^\text{?}H(X,Y |Z) \\ &= H(X|Z) + H(Y|X, Z)\\ &\ge H(X|Z) \end{aligned} \end{equation}

Can someone explain to me where the first inequality and identity come from. Why would adding a another random variable decrease entropy in the first inequality?

Answer 1

The first inequality means that in general conditioning may reduce the information. If you prefer, note that $$H(X)\ge H(X|Z)$$ and then condition on $Z$ on both sides. For the identity, similarly, since $$H(X|Y)+H(Y)=H(X,Y),$$ after conditioning on $Z$ on both sides you get the inequality above.