For any $n\ge2$ prove that $H(X_1,X_2,...,X_n)\ge\sum\limits_{i=n}^\mathbb{n}\ H(X_i|X_j , j \neq i)$

Question

I am trying to figure this out and I am stuck. Any ideas?

For any $n\ge2$ prove that $H(X_1,X_2,\ldots,X_n)\ge\sum\limits_{i=1}^\mathbb{n}\ H(X_i\mid X_j , \ j \neq i)$

Answer 1

For n=2 we have $$H(X1,X2)=H(X1)+H(X2|X1) \ge H(X_1|X_2)+ H(X_2|X_1) \ (1)$$ which stands because conditioning does not increase entropy.

Same logic for n>2

Answer 2

Hint: Use the chain rule $$ H(X_1,\ldots,X_n) = H(X_1) + H(X_2|X_1) + H(X_3|X_2,X_1) + \cdots + H(X_n|X_{n-1},\ldots,X_1). $$