Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy

This study focuses on conditional entropy frameworks based on the Kolmogorov–Nagumo (KN) mean. First, $(\eta ,\psi )$-KN averaging ($\mathtt{\text{EPKNAVG}}$), a KN-mean extension of the $\eta$-averaging ($\mathtt{\text{EAVG}}$) framework for $(\eta ,F)$-entropies, is introduced and proven to be equivalent to $\mathtt{\text{EAVG}}$ under suitable concavification conditions. Second, motivated by generalized $g$-vulnerability, a new framework is proposed for generalized $g$-conditional entropies. This framework captures conditional entropies beyond the scope of $\mathtt{\text{EAVG}}$-type representations. In particular, it is shown that there exists an $\alpha$ and a joint probability distribution $p_{X,Y}$ such that the Augustin–Csisz{' a}r conditional entropy $H_{\alpha }^{\mathrm{C}}(X\mathrm{\mid }Y)$ cannot be represented by any $(\eta ,F)$-entropy satisfying $\mathtt{\text{EAVG}}$. In contrast, it is represented within the proposed framework. Furthermore, sufficient conditions are derived under which the proposed generalized $g$-conditional entropies satisfy the conditioning reduces entropy property and the data-processing inequality.