Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy

@misc{kamatsuka2026kolmogorovnagumo,
  title = {{Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy}},
  author = {Akira Kamatsuka and Takahiro Yoshida},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2605.07624},
  eprint = {2605.07624},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.07624},
}

TY  - UNPB
TI  - Kolmogorov--Nagumo Mean Frameworks for Conditional Entropy
AU  - Akira Kamatsuka
AU  - Takahiro Yoshida
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2605.07624
UR  - https://arxiv.org/abs/2605.07624
AB  - 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^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.
ER  -