Kullback–Leibler Asymmetry and Information Homogeneity in Natural Exponential Families
- 1Faculty of Industrial Engineering and Technology Management, HIT—Holon Institute of Technology, Holon 5810201, Israel
Abstract
Natural exponential families admit two complementary structural descriptions: through their variance functions and through the information geometry generated by relative entropy and Fisher information. This paper establishes a direct connection between these descriptions. For a regular one-dimensional natural exponential family, we show that a fixed global direction of Kullback–Leibler asymmetry is equivalent to monotonicity of the variance function and, equivalently, to a fixed sign of the standardized skewness. In fact, a local ordering condition around each parameter point already forces the same ordering globally. We then reparametrize the family by Fisher–Rao arc length, which coincides with the classical variance-stabilizing coordinate, and study local spatial homogeneity of information divergence. Local homogeneity of the directed Kullback–Leibler divergence, of its antisymmetric part, of the magnitude of that asymmetry, or local radiality of Jeffreys divergence are shown to be equivalent. Each condition forces constant standardized skewness and consequently characterizes, up to a nonsingular affine transformation, precisely the normal location and fixed-shape gamma natural exponential families. Thus, weak local information conditions determine the global structure of the family. We further show that the global conclusions already follow from a sequential local ordering in the first theorem, and that position-independence of either the leading cubic KL-asymmetry coefficient or the quartic Jeffreys-excess coefficient alone yields the same normal/gamma rigidity.
Related
- Equivalence on exponential families as Hessian manifolds and classification of exponential families of order 1 with constant Hessian sectional curvature
- Cramer-Rao Type Bounds for Generalized Exponential Families
- On Natural Exponential Families with Log-Affine Generalized Variance
- Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families
- Generalized Einstein Laurent polynomials, Toric Kähler-Einstein Rigidity, and Finite Exponential Families
BibTeX
@article{barlev2026kullbackleibler,
title = {{Kullback–Leibler Asymmetry and Information Homogeneity in Natural Exponential Families}},
author = {Shaul K. Bar-Lev},
journal = {Mathematics},
year = {2026},
month = nov,
doi = {10.3390/math14213992},
url = {https://www.mdpi.com/2227-7390/14/21/3992},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Kullback–Leibler Asymmetry and Information Homogeneity in Natural Exponential Families
AU - Shaul K. Bar-Lev
PY - 2026
DA - 2026/11/02
JO - Mathematics
DO - 10.3390/math14213992
UR - https://www.mdpi.com/2227-7390/14/21/3992
AB - Natural exponential families admit two complementary structural descriptions: through their variance functions and through the information geometry generated by relative entropy and Fisher information. This paper establishes a direct connection between these descriptions. For a regular one-dimensional natural exponential family, we show that a fixed global direction of Kullback–Leibler asymmetry is equivalent to monotonicity of the variance function and, equivalently, to a fixed sign of the standardized skewness. In fact, a local ordering condition around each parameter point already forces the same ordering globally. We then reparametrize the family by Fisher–Rao arc length, which coincides with the classical variance-stabilizing coordinate, and study local spatial homogeneity of information divergence. Local homogeneity of the directed Kullback–Leibler divergence, of its antisymmetric part, of the magnitude of that asymmetry, or local radiality of Jeffreys divergence are shown to be equivalent. Each condition forces constant standardized skewness and consequently characterizes, up to a nonsingular affine transformation, precisely the normal location and fixed-shape gamma natural exponential families. Thus, weak local information conditions determine the global structure of the family. We further show that the global conclusions already follow from a sequential local ordering in the first theorem, and that position-independence of either the leading cubic KL-asymmetry coefficient or the quartic Jeffreys-excess coefficient alone yields the same normal/gamma rigidity.
SN - 2227-7390
ER -