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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2605.07624 |
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| _version_ | 1866917482931945472 |
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| author | Kamatsuka, Akira Yoshida, Takahiro |
| author_facet | Kamatsuka, Akira Yoshida, Takahiro |
| contents | This study focuses on conditional entropy frameworks based on the Kolmogorov--Nagumo (KN) mean. First, $(η, ψ)$-KN averaging (\texttt{EPKNAVG}), a KN-mean extension of the $η$-averaging (\texttt{EAVG}) framework for $(η, F)$-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 $α$ and a joint probability distribution $p_{X, Y}$ 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}. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07624 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy Kamatsuka, Akira Yoshida, Takahiro Information Theory This study focuses on conditional entropy frameworks based on the Kolmogorov--Nagumo (KN) mean. First, $(η, ψ)$-KN averaging (\texttt{EPKNAVG}), a KN-mean extension of the $η$-averaging (\texttt{EAVG}) framework for $(η, F)$-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 $α$ and a joint probability distribution $p_{X, Y}$ 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}. 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. |
| title | Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy |
| topic | Information Theory |
| url | https://arxiv.org/abs/2605.07624 |