Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have very hard time to prove the following inequality or to show a contradiction.

$H(X_1,X_2,X_3) + H(X_1,X_2,X_4)+ H(X_1,X_3,X_4) + H(X_2,X_3,X_4) \leq 3(H(X_1,X_2) + H(X_3,X_4))$

The problem is I don't know how to approach the solution.

I would appreciate for any help.

share|improve this question
    
Are $X_1, X_2...$ random variables? –  Henrique Oct 10 '13 at 15:27

1 Answer 1

Hint:

Note that $H(X,Y) \leq H(X) + H(Y)$.

$$ H(X_1, X_2, X_3) \leq H(X_1, X_2) + H(X_3)\\ H(X_1, X_2, X_4) \leq H(X_1, X_2) + H(X_4)\\ H(X_1, X_3, X_4) \leq H(X_3, X_4) + H(X_1)\\ H(X_2, X_3, X_4) \leq H(X_3, X_4) + H(X_2) $$ Therefore, $$ LHS \leq 2(H(X_1, H_2) + H(X_3, H_4)) + H(X_1) + H(X_2) + H(X_3) + H(X_4) $$

Can you find a counter example now?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.