Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy
This work addresses a theoretical gap in information theory by providing a more general representation for conditional entropies, but it is incremental as it extends existing frameworks.
The paper introduces a new framework for conditional entropy based on the Kolmogorov-Nagumo mean, which captures entropies like Augustin-Csiszár conditional entropy that cannot be represented by existing averaging frameworks. It provides sufficient conditions for the new framework to satisfy properties like conditioning reduces entropy and data-processing inequality.
We study conditional entropy frameworks based on the Kolmogorov--Nagumo (KN) mean. First, we introduce $(η, ψ)$-KN averaging (\texttt{EPKNAVG}), a KN-mean extension of the $η$-averaging (\texttt{EAVG}) framework for $(η, F)$-entropy, and prove that, under suitable concavification conditions, it is equivalent to \texttt{EAVG}. Second, motivated by generalized $g$-vulnerability, we propose a new framework of generalized $g$-conditional entropies. We show that this framework captures conditional entropies beyond the scope of \texttt{EAVG}-type representations. In particular, there exists $α\in(0,1)\cup(1,\infty)$ such that the Augustin--Csisz{\' a}r conditional entropy $H_α^{\mathrm{C}}(X|Y)$ cannot be represented by any $(η,F)$-entropy satisfying \texttt{EAVG}, whereas it is represented within the proposed framework. We further derive sufficient conditions for the proposed generalized $g$-conditional entropies to satisfy conditioning reduces entropy (\texttt{CRE}) and the data-processing inequality (\texttt{DPI}).