Let $p(\cdot | \cdot)$ and $q(\cdot | \cdot)$ be two known channels from alphabet $X$ to $Y$. What is the optimal strategy to distinguish between them if you are allowed to use the channel only once, and what is the corresponding success probability? (Note that you can choose which probability distribution you want to use as input).
My thoughts: Since we can choose which probability distribution we want to use as input, we'll choose one with $P(x)=1$ $(x \in X)$. If $x$ goes through a channel ($p$ or $q$), and what comes out is $y \in Y$, then we will guess the channel was $p$ if $p(y|x)>q(y|x)$ and we will guess $q$ otherwise. I have a feeling that the success probability will be related to the statistical distance $\delta (p,q)=\frac{1}{2}\sum\limits_{y}|p(y|x)-q(y|x)|$. Am I correct? It makes sense that if probability distributions $p$ and $q$ are very different, the probability of guessing correctly will be large, just like the statistical distance. But how is it related?
Any help and comments would be much appreciated.