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}).

翻译：我们研究基于Kolmogorov--Nagumo（KN）均值的条件熵框架。首先，我们引入$(η, ψ)$-KN平均（\texttt{EPKNAVG}），这是针对$(η, F)$-熵的$η$-平均（\texttt{EAVG}）框架的KN均值推广，并证明在适当的凹性化条件下，它与\texttt{EAVG}等价。其次，受广义$g$-脆弱性概念的启发，我们提出一个广义$g$-条件熵的新框架。我们证明该框架能够刻画超出\texttt{EAVG}型表示范围的各类条件熵。特别地，存在$α\in(0,1)\cup(1,\infty)$使得Augustin--Csiszár条件熵$H_α^{\mathrm{C}}(X|Y)$无法被任何满足\texttt{EAVG}的$(η,F)$-熵所表示，但能在所提框架内实现表示。我们进一步推导了所提出的广义$g$-条件熵满足“条件作用降低熵”（\texttt{CRE}）和“数据处理不等式”（\texttt{DPI}）的充分条件。