Conditional mutual-information chain rule and data processing for standard-Borel variables

For $P_S$-almost every $s$, apply [F2] to the conditional law of $(U,V,Y)$ given $S=s$. It gives $I((U,V);Y\mathrm{\mid }S=s)=I(U;Y\mathrm{\mid }S=s)+I(V;Y\mathrm{\mid }U,S=s)\mathrm{.}$ Integrate over $P_S$; by the conditional-KL definition [F3] and disintegration chain rule [F4], the three integrals are the corresponding conditional mutual informations given $S$. This proves the first identity without subtracting extended values.

Exchange the roles of $U$ and $V$ in step 1.1 to obtain the second identity. The symmetry of conditional mutual information between the grouped variables follows from [F6].

Suppose the conditional law of $Y$ given $(U,V,S)$ depends only on $(V,S)$. Then $U$ and $Y$ are conditionally independent given $(V,S)$, so [F3] gives $I(U;Y\mathrm{\mid }V,S)=0$. The two chain-rule identities in steps 1.1 and 2.1 now give $I(U;Y\mathrm{\mid }S)+I(V;Y\mathrm{\mid }U,S)=I(V;Y\mathrm{\mid }S)$. Since $I(V;Y\mathrm{\mid }U,S)\ge 0$ by [F5], the claimed inequality follows.