Information geometry of emergent symmetry quotients and their weak unfoldings

Abstract

A regular observed statistical model may converge to a limit in which a previously identifiable signed parameter becomes identifiable only modulo a reflection. We study the local information geometry of this transition. For a twice differentiable Hellinger embedding with an exact limiting reflection, the observed displacement is forced into the two-jet form \(\varepsilon\lambda J_-+\lambda^2J_+/2\), up to higher-order terms. The mixed jet restores the sign away from the symmetric face, whereas the even jet is the first tangent inherited by the quotient. After nuisance elimination, a positive Gram determinant yields a nondegenerate cross-cap two-jet. The associated local asymptotic theory has three regimes governed by \(\tau_n=\sqrt n\,\varepsilon_n^2\): regular signed LAN, a critical curved Gaussian subexperiment, and a quotient regime with the \(n^{-1/4}\) signed scale. We prove that the same parabolic critical experiment persists for predictive likelihoods along a single stationary dependent trajectory. The limiting quotient has a regular Fisher metric in the invariant coordinate, while its pullback degenerates in the signed coordinate. For a solvable CIR--OU benchmark motivated by coherent sea-clutter observations, we derive the quotient Fisher metric and curvature explicitly and show that the curvature is strictly negative. We also determine the restricted holonomy of the full Amari family: \(\operatorname{Hol}_0(\nabla^{(a)})=SO(2)\) for \(a=0\), whereas \(\operatorname{Hol}_0(\nabla^{(a)})=GL^+(2,\mathbb R)\) for \(a\neq0\). The results separate the intrinsic geometry of the limiting quotient from the transverse geometry of its weak unfolding.