# Deriving a lower bound on the conditional entropy, conditioned on an event

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 $$\sum_w \Pr[W=w]\ H[ Z \mid W=w,\mathcal{E}]$$?
• Fixed the reference to the lemma in the paper. Jun 22 at 7:49

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 $$H(Z|W) \leq H(Z,\mathcal{E}|W) = H(Z|\mathcal{E},W) + H(\mathcal{E}|W) \leq H(Z|\mathcal{E},W) + H(\mathcal{E}).$$