# Auxiliary random variables in the analysis of the private information of wiretap channels

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!