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bugfix
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Trance-0
2025-11-22 15:18:52 -06:00
parent 87f31f5c7f
commit 000922f9bd
4 changed files with 134 additions and 6 deletions
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content/CSE5313/CSE5313_L18.md
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@@ -196,7 +196,7 @@ $\operatorname{Pr}(s_\mathcal{Z}|m_1, \cdots, m_{t-z}) = \operatorname{Pr}(U_1,
Conclude similarly by the law of total probability.
$\operatorname{Pr}(s_\mathcal{Z}|m_1, \cdots, m_{t-z}) = \operatorname{Pr}(s_\mathcal{Z}) \implies I(S_\mathcal{Z}; M_1, \cdots, M_{t-z}) = 0.
$\operatorname{Pr}(s_\mathcal{Z}|m_1, \cdots, m_{t-z}) = \operatorname{Pr}(s_\mathcal{Z}) \implies I(S_\mathcal{Z}; M_1, \cdots, M_{t-z}) = 0$.
### Conditional mutual information
@@ -246,14 +246,14 @@ A: Fix any $\mathcal{T} = \{i_1, \cdots, i_t\} \subseteq [n]$ of size $t$, and l
$$
\begin{aligned}
H(M) &= I(M; S_\mathcal{T}) + H(M|S_\mathcal{T}) \text{(by def. of mutual information)}\\
&= I(M; S_\mathcal{T}) \text{(since S_\mathcal{T} suffice to decode M)}\\
&= I(M; S_{i_t}, S_\mathcal{Z}) \text{(since S_\mathcal{T} = S_\mathcal{Z} ∪ S_{i_t})}\\
&= I(M; S_\mathcal{T}) \text{(since }S_\mathcal{T}\text{ suffice to decode M)}\\
&= I(M; S_{i_t}, S_\mathcal{Z}) \text{(since }S_\mathcal{T} = S_\mathcal{Z} ∪ S_{i_t})\\
&= I(M; S_{i_t}|S_\mathcal{Z}) + I(M; S_\mathcal{Z}) \text{(chain rule)}\\
&= I(M; S_{i_t}|S_\mathcal{Z}) \text{(since \mathcal{Z} ≤ z, it reveals nothing about M)}\\
&= I(M; S_{i_t}|S_\mathcal{Z}) \text{(since }\mathcal{Z}\leq z \text{, it reveals nothing about M)}\\
&= I(S_{i_t}; M|S_\mathcal{Z}) \text{(symmetry of mutual information)}\\
&= H(S_{i_t}|S_\mathcal{Z}) - H(S_{i_t}|M,S_\mathcal{Z}) \text{(def. of conditional mutual information)}\\
\leq H(S_{i_t}|S_\mathcal{Z}) \text{(entropy is non-negative)}\\
\leq H(S_{i_t}|S_\mathcal{Z}) \text{(conditioning reduces entropy). \\
&\leq H(S_{i_t}|S_\mathcal{Z}) \text{(entropy is non-negative)}\\
&\leq H(S_{i_t}|S_\mathcal{Z}) \text{(conditioning reduces entropy)} \\
\end{aligned}
$$