If $H(X,Y) = H(X) + H(Y)$ and $H(Y,Z) = H(Y) + H(Z)$, is $H(X,Y,Z) = H(X) + H(Y) + H(Z)$?

Question

The question in the title. $H(X)$ refers to the entropy of $X$.

Given 3 R.V., if they satisfy $H(X,Y) = H(X) + H(Y)$ and $H(Y,Z) = H(Y) + H(Z)$, does that imply that $H(X,Y,Z) = H(X) + H(Y) + H(Z)$.

My guess is that no it isn't true, but I can't think of a counterexample.

Answer 1

No.

Counterexample: let $X=Z$, and let $Y$ be any random variable that is independent of $X$ and $Z$. Then you have:

• $H(X, Y)=H(X)+H(Y)$

• $H(Y, Z)=H(Y)+H(Z)$

• $H(X, Y, Z)=H(X)+H(Y)\ne H(X)+H(Y)+H(Z)$