Chain rule for conditional versions of Sibson’s α-mutual information

Ascertain whether any conditional Sibson α-mutual information definition introduced in Section 6—such as I^{Y|Z}_α(X,Y|Z)=min_{Q_{Y|Z}} D_α(P_{XYZ}||P_{X|Z}Q_{Y|Z}P_Z) or I^{Z}_α(X,Y|Z)=min_{Q_Z} D_α(P_{XYZ}||P_{X|Z}P_{Y|Z}Q_Z)—satisfies a Mutual Information-like chain rule of the form I_α(X;(Y,Z)) ≤ I_α(X;Y) + I^{•}_α(X;Y|Z), where I^{•}_α denotes one of these conditional definitions. Establish such a chain rule or provide counterexamples if it fails.

Background

The authors present multiple conditional generalizations of Sibson’s α-mutual information (e.g., minimizing Rènyi divergence over Q_{Y|Z} or Q_Z) and derive closed-form expressions. They verify that these measures recover Shannon’s conditional mutual information in the limit α→1 and connect to conditional maximal leakage for α→∞. However, it remains unclear whether a chain rule analogous to Shannon’s—relating the joint dependence I_α(X;(Y,Z)) to the marginal and conditional terms—holds for any of the proposed definitions.

This open question targets a fundamental structural property of conditional α-information measures and, if resolved, could clarify their suitability as operational analogues to mutual information in settings involving side information or conditioning.