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Imagine a Binary Erasure Channel as depicted on Wikipedia.

One equation describing the mutual information is:

$$ \begin{align*}I(x;y) &= H(x) - H(x|y) \\ &= H(x) - p(y=0) \cdot 0 - p(y=?) \cdot H(x) -p(y=1)\cdot 0\end{align*}$$

Why is it "$p(y=?) \cdot H(x)$" and not "$p(y=?) \cdot H(x|y=?)$"?

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If $y = ?$ then $H(x) = H(x|y=?)$ since $y = ?$ gives you no information about $x$.

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  • $\begingroup$ I see, is that because p(x=0|y=?)=p(x=1|y=?) in a BEC? $\endgroup$ – user38931 May 6 '14 at 19:32
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    $\begingroup$ @user38931 The identity $H(x|y=?) = H(x)$ would be true even if $x$ were skewed. What we need is $p(x=0|y=?) = p(x=0)$. $\endgroup$ – Yuval Filmus May 6 '14 at 19:44

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