Statistical Bergman geometry

Abstract

This paper explores the Bergman geometry of bounded domains $\Omega$ in $\mathbb{C}^n$ through the lens of information geometry by introducing a mapping $\Phi: \Omega \rightarrow \mathcal{P}(\Omega)$, where $\mathcal{P}(\Omega)$ denotes a space of probability measures on $\Omega$. A result by J. Burbea and C. Rao establishes that the pullback of the Fisher information metric, the fundamental Riemannian pseudo-metric in information geometry, via $\Phi$ coincides with the Bergman metric of $\Omega$. Building on this idea, we consider $\Omega$ as a statistical model and present several interesting results within this framework. First, we derive a new statistical curvature formula for the Bergman metric by expressing it in terms of covariance. Second, given a proper holomorphic map $f: \Omega_1 \rightarrow \Omega_2$, we prove that if the induced measure push-forward $\kappa: \mathcal{P}(\Omega_1) \rightarrow \mathcal{P}(\Omega_2)$ preserves the Fisher information metrics, then $f$ must be a biholomorphism. Finally, we establish the consistency and the central limit theorem of the Fr\'echet sample mean for Calabi's diastasis function.