Soft summaries, output bias, and asymmetric noise

For a soft $u$, the conditional entropy $\mathbb{E}h(u)$ need not vanish. Replacing $\mathcal{I}(u)$ by $h(\mathbb{E}u)$ would erase that randomization cost and change the fixed-information problem.

Arbitrary output bias means that $m=\mathbb{E}f$ is unrestricted while $X$ remains uniform. It does not mean that the input distribution is arbitrary.

Asymmetric bit transitions generally destroy the symmetry $K(x,y)=K(y,x)$, uniform stationarity, and the one-parameter Walsh multiplier ${\rho }^{|A|}$. Such a model requires a new posterior kernel and a new comparison theorem.