I am following Section 13.2 of Mark Wilde's book. I reproduce the question here for completeness.

Consider a wiretap channel $X\rightarrow Y,Z$ defined by the conditional probabilities $p(y,z|x)$ for inputs chosen according to some $p(x)$. The goal of maximizing private information is to transmit information from $X$ to $Y$ without any information leaking to $Z$. Naively, one would say this is simply

$$\max_{p_{X}(x)}I(X:Y) - I(X:Z),$$

where for any channel $A\rightarrow B$ with input chosen according $p(a)$ and symbols transmitted according to $p(b|a)$, $I(A:B)$ is the mutual information between $A$ and $B$ and given by

$$I(A:B) = \sum_{a,b} p(a,b)\log\left(\frac{p(a,b)}{p(a)p(b)}\right)$$

However, the correct way to think about this is to introduce an auxiliary variable $U$. Alice (the sender at $X$) decides $x\in X$ based on $u\in U$. That is $p(x) = \sum_u p(x|u)p(u)$. The private information is defined as

$$\max_{p_{U,X}(u,x)}[I(U:Y) - I(U:Z)]$$

The question is why one needs to introduce the auxiliary variable? I don't really have either intuition or a mathematical proof so any pointers are appreciated!


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