Discussion Forum

Hello,

Let $X,Y,Z$ be discrete random variables on $\Omega,\mathcal{F},\Bbb{P}$ a probability space. Then is it true that $H(X,Y,Z)=H(X)+H(Y,Z|X)$ ? I have two ideas to prove this result. The first uses the chain rule : $H(X_1,X_2,X_3)=H(X_1)+H(X_2|X_1)+H(X_3|X_2,X_1)$ and since (not sure of this) $H(X_2|X_1)+H(X_3|X_2,X_1)=H(X_2,X_3|X_1)$ we have the result. The second uses the definition of conditional entropy :

$H(X)+H(Y,Z|X)=-\sum_xp(x)\log[p(x)]-\sum_{x,y,z}p(x,y,z)\log[p(y,z | x)]$

$=-\sum_{x,y,z}p(x,y,z)\log[p(x)]-\sum_{x,y,z}p(x,y,z)\log[p(y,z | x)$

$=-\sum_{x,y,z}p(x,y,z)\log[\underbrace{p(x)p(y,z | x)}_{p(x,y,z)}]=H(X,Y,Z)$

Are my two "proofs" correct ? Best regards, GL