Nonexistence of dual measures of the NEF-GHS

This paper studies the inverse problem in information geometry: given a Hessian manifold, is it derived from an open exponential family? As a motivating case, we prove that for the natural exponential family generated by the generalized hyperbolic secant distribution (NEF-GHS) with shape parameter $$p>0$$ p > 0 , the dual Hessian manifold is not derived from any open exponential family, thereby extending a result of Letac for the case $$p=1$$ p = 1 . Our proof exploits a local obstruction coming from branch points in the Hessian potential. Inspired by this observation, we prove a broad result: when a Hessian metric admits a real-analytic global Hessian potential, “derivedness” is determined by its restrictions to open submanifolds.