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I came across Lemma 19 in Certifying Equality With Limited Interaction, which states the following for jointly distributed random variables $Z$, $W$, where $Z$ takes values in $\{0,1\}^n$, and some arbitrary event $\mathcal{E}$: $$ H[ Z \mid W ] \ge n - d\ \Longrightarrow\ H[ Z \mid W, \mathcal{E}] \ge n - \frac{d+1}{\Pr[\mathcal{E}]}. $$

The proof is very short, but I'm unable to follow one of the derivations, which seems to be using a general property of conditional entropy that I'm unfamiliar with.

Proof of Lemma 19: We know that the entropy of $Z$ can be at most $n$ no matter on what we condition. Let $H_b$ denote the binary entropy. It follows that $$ \begin{align} n -d &\le H[ Z \mid W] \\ &= \Pr[ \mathcal{E} ] H[Z \mid W, \mathcal{E}] + \Pr[\neg \mathcal{E}]\ H[Z \mid W, \neg \mathcal{E}] + H_b(\Pr[\mathcal{E}]) \quad (2)\\ &\le \Pr[ \mathcal{E}]\ H[Z \mid W, \mathcal{E}] + \Pr[\neg \mathcal{E}]\cdot n + 1. \end{align} $$ QED

My Questions:

  1. Why does (2) hold? Is this some general property of conditional entropy?
  2. How exactly is $H[ Z \mid W, \mathcal{E} ]$ defined? Is it equivalent to $\sum_w \Pr[W=w\mid \mathcal{E}]\ H[ Z \mid W=w,\mathcal{E}]$ or $\sum_w \Pr[W=w]\ H[ Z \mid W=w,\mathcal{E}]$?
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  • $\begingroup$ Fixed the reference to the lemma in the paper. $\endgroup$
    – New_In_CS
    Jun 22 at 7:49
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We treat $\mathcal{E}$ as an indicator random variable for the event $\mathcal{E}$ (so $\mathcal{E}$ stands for two different things). The derivation is $$ H(Z|W) \leq H(Z,\mathcal{E}|W) = H(Z|\mathcal{E},W) + H(\mathcal{E}|W) \leq H(Z|\mathcal{E},W) + H(\mathcal{E}). $$

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