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I have two questions both related to Shearer's Inequality:

1) When is equality attained in Shearer's Inequality? One trivial instance is when the random variables are all independent. Is this the only instance?

2) This is an exercise problem for which hints would be appreciated: Show that for any random variables $X_1, X_2, X_3, X_4 $,

$$H(X_1, X_2, X_3) + H(X_2, X_3, X_4) + H(X_3,X_4,X_1) + H(X_4, X_1, X_2) \leq \frac{3}{2}(H(X_1, X_2) + H(X_2, X_3) + H(X_3,X_4) + H(X_4, X_1)) $$

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@Raskolnikov: Thanks so much for the edit. I wasn't sure what the problem was and kept struggling to fix it. – BharatRam Jul 16 '13 at 14:34
There was an underscore too much. – Raskolnikov Jul 16 '13 at 14:42
I managed the second question by conditional entropies. The first question still stands. – BharatRam Jul 17 '13 at 3:50

HINT: Apply Shearer's inequality to each term of the left hand side of the inequality separately. Als apply Shearer's inequality to $H(X_1,X_2,X_3,X_4)$ by reducing to entropies dependent on three RVs and on two RVs. Note you can do the latter in 3 different ways. Use your result from 1.

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