Show that $ H(X,Y) + H(Y,Z) + H(X,Z) \geq 2H(X,Y,Z)$

Question

Let $H$ be the entropy function and $X,Y,Z$ random variables, show that $ H(X,Y) + H(Y,Z) + H(X,Z) \geq 2 H(X,Y,Z)$

I have tried to use some identities like $H(U,V) = H(U) + H(V|U)$ then

$$H(X,Y,Z) = H(X,Y)+ H(Z|X,Y)$$

$$H(X,Y,Z) = H(X,Z) + H(Y|X,Z)$$

Then I should show that $H(Z|X,Y) + H(Y|X,Z) \leq H(Y,Z)$ but I don't know how to proceed, any idea would be very appreciated.

Thanks

Answer 1

Note that $$ \begin{align} H(Y,Z) &\geq H(Y,Z|X)\\ &= H(Y|Z,X) + H(Z|X)\\ &\geq H(Y|Z,X) + H(Z|X,Y), \end{align} $$ with the inequalities following due to conditioning on a random variable can only reduce entropy