Statistical Bergman geometry
Gunhee Cho, Jihun Yum
TL;DR
This work builds a bridge between Bergman geometry and Information geometry by embedding a bounded domain into the probability simplex via Φ(z)=P(z,ξ)dV(ξ) with P(z,ξ)=|ℬ(z,ξ)|^2/ℬ(z,z), showing that the pullback of the Fisher information metric equals the Bergman metric: (Φ^* g_F)=g_B. It derives a covariance-based statistical curvature formula for g_B, and proves a sufficiency criterion: if the push-forward under a holomorphic map preserves Fisher information, then the map is biholomorphic; it also establishes a Fréchet mean consistency and central limit theorem using Calabi’s diastasis in the Bergman metric. The results highlight monotonicity properties of Fisher information under measure push-forward and provide a probabilistic-informational lens on Bergman geometry with potential implications for complex analysis and geometric statistics.
Abstract
This paper explores the Bergman geometry of bounded domains $Ω$ in $\mathbb{C}^n$ through the lens of Information geometry by introducing an embedding $Φ: Ω\rightarrow \mathcal{P}(Ω)$, where $\mathcal{P}(Ω)$ denotes a space of probability distributions on $Ω$. A result by J.Burbea and C. Rao establishes that the pullback of the Fisher information metric, the fundamental Riemannian metric in Information geometry, via $Φ$ coincides with the Bergman metric of $Ω$. Building on this idea, we consider $Ω$ as a statistical model in $\mathcal{P}(Ω)$ and present several interesting results within this framework. First, we drive a new statistical curvature formula for the Bergman metric by expressing it in terms of covariance. Second, given a proper holomorphic map $f: Ω_1 \rightarrow Ω_2$, we prove that if the measure push-forward $κ: \mathcal{P}(Ω_1) \rightarrow \mathcal{P}(Ω_2)$ of $f$ preserves the Fisher information metrics, then $f$ must be a biholomorphism. Finally, we establish consistency and the central limit theorem of the Fréchet sample mean for the Calabi's diastasis function.