Intrinsic geometric exponential calculus

Construct and establish the admissibility of intrinsic chart-based logarithm and exponential operations, \(\log_G\) and \(\exp_G\), that satisfy algebraic consistency, normalization compatibility, reparametrization invariance, and a coherent statistical interpretation, thereby determining whether probability geometry yields an exponential calculus broader than transported G-exponential families.

Background

The paper’s construction transports an ordinary classical exponential family from the unit interval through a probability-coordinate chart. It therefore changes the geometry in which exponential structure is represented without changing the exponential function itself.

The authors identify a deeper unresolved program: defining logarithm and exponential operations intrinsically from the probability geometry rather than importing the ordinary exponential through coordinates. Any such construction would have to meet stringent algebraic, normalization, invariance, and statistical requirements, and the paper does not provide a canonical candidate.

References

Finally, the intrinsic question of Section~\ref{sec:intrinsic}---whether probability geometry generates its own exponential calculus---remains, in our view, the deepest open direction, and the transported theory developed here is intended as its rigorous point of departure.

— G-Exponential Families through Probability Coordinates  (2609.03208 - Cojocea, 2 Sep 2026) in Section 7, “Towards intrinsic geometric exponentiality”; reiterated in Section 8, “Discussion and outlook”

Several directions remain open. The anchoring caveat of Section~\ref{sec:inference} calls for an estimated-chart extension: a two-step theory in the sense of in which the chart family $G_{\mu,\sigma}=G_0((\cdot-\mu)/\sigma)$ is calibrated before the exponential parameter is estimated, restoring equivariance at the price of a calibration correction. Chart selection---which benchmark geometry to impose, and how to compare fits across charts given the information invariance of Proposition~\ref{prop:invariance}---is a model-choice problem specific to this framework. Multivariate extensions through the copula coordinates of would transport exponential families on the unit cube.

— G-Exponential Families through Probability Coordinates  (2609.03208 - Cojocea, 2 Sep 2026) in Section 8, “Discussion and outlook”

In information geometry, mixture coordinates and exponential coordinates are dual to each other. It is interesting to see in what sense the language constructs of $\mathsf{score}$ and $\mathsf{if}$ could be said to be dual. We do not have a good answer, and leave that for future work, but note that one interesting angle is through~\citep{richardson2024mixture}.

— Compositional Information Geometry of Probabilistic Programs  (2610.17886 - Karwowski et al., 13 Oct 2026) in Section 4, subsection "Normalisation", final remark