Abstract:This study focuses on conditional entropy frameworks based on the Kolmogorov--Nagumo (KN) mean. First, $(\eta, \psi)$-KN averaging (\texttt{EPKNAVG}), a KN-mean extension of the $\eta$-averaging (\texttt{EAVG}) framework for $(\eta, F)$-entropies, is introduced and proven to be equivalent to \texttt{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 \texttt{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 $H_{\alpha}^{\mathrm{C}}(X|Y)$ cannot be represented by any $(\eta,F)$-entropy satisfying \texttt{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.
From: Akira Kamatsuka [view email]
[v1]
Fri, 8 May 2026 11:53:49 UTC (81 KB)
[v2]
Mon, 11 May 2026 11:07:22 UTC (81 KB)
[v3]
Tue, 12 May 2026 10:03:58 UTC (81 KB)
[v4]
Mon, 31 Aug 2026 13:37:49 UTC (83 KB)
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