Does the "information can't hurt" inequality for conditional entropy $H(X)\ge H(X\mid Y)$ extend to $H(X\mid Y)\ge H(X\mid Y,Z)$?
2026-03-25 18:59:48.1774465188
conditional entropy inequality
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Yes, of course. That's a consequence of mutual information being non-negative also if conditioned on other variable: $I(X;Z | Y) = H(X | Y) - H(X | Y,Z) \ge 0$
BTW: the "information can't hurt" motto is basically right, but it can be wrongly understood. It might lead to you conclude that $I(R;S|T)\ge I(R;S )$ (knowledge about $T$ can't reduce the mutual information between $R,S$ ... right? wrong, it can).