I have a question regarding the capacity of a channel
Consider a channel given by the transition probabilities $p(y|x)$ with capacity $C$. Now a friendly statistician offers to preprocess the output for you by applying a function $g$, giving $\bar{y} = g(y)$. Show that his strategy can only decrease the capacity of the channel, and under which conditions on $g$ is the capacity strictly decreased?
This is my answer for the 1st part.
Since $X\rightarrow Y \rightarrow \bar{Y}$. We can apply the data processing inequality, $I(X;Y)\ge I(X;\bar{Y})$. Let $\bar{p}(x)$ be the distribution maximising $I(X;\bar{Y})$, then $$C=\max_{p(x)}I(X;Y)\geq I(X;Y)_{p(x)=\bar{p}(x)}\geq I(X;\bar{Y})_{p(x)=\bar{p}(x)}=\max_{p(x)}I(X;\bar{Y})=\bar{C}$$ Hence the capacity can never increase the capacity of the channel.
However, how do I do the second part? Thanks in advance.