Chain rule for conditional entropy ($H(X,Z\mid Y)=H(X\mid Y)+H(Z\mid X,Y)$)

For classical finite random variables, $H(X,Z\mid Y)=H(X\mid Y)+H(Z\mid X,Y)=H(Z\mid Y)+H(X\mid Z,Y)$. Each identity follows by inserting the corresponding joint entropy into $H(X,Z,Y)-H(Y)$. The two orders are particularly useful when one variable is a deterministic function of the others, as in Fano's inequality via an error indicator.