Bit-position information in a binary channel ($I(X;Y\mid J)-I(X;Y)=I(X;J\mid Y)\ge 0,X\perp J$)

Suppose a fair input bit $X$ is independent of a publicly known channel-position label $J$. At position $j$, let the channel be a binary symmetric channel with error probability $p_j$. Forgetting $J$ gives a binary symmetric channel with error $\overline{p}=\mathbb{E}{p}_J$. ThereforeThe chain rule for mutual information and $I(X;J)=0$ show that their difference is the nonnegative conditional mutual information $I(X;J\mid Y)$. Equivalently, the inequality follows from concavity of binary entropy. For $p_0=1/5,p_1=0$ with equal position weights, the known-position value is $0.639036$ bits and the pooled value is $0.531004$ bits. These quantities answer different statistical questions.