Problem

Given a DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$, determine the Information Capacity $$C=\max_{p_{X}}I(X;Y)$$First note that necessary and sufficient conditions for finding $C$ can be shown since $I(X;Y)=I(p_{X},p_{Y|X})$ is concave in $p_{X}$; but they might be hard to use as they require “guessing” a priori the optimal input distribution.

$\therefore$ In general, $C$ does not admit a closed-form expression (in terms of channel parameters), unless if the channel exhibits certain “symmetry” properties.