Jeffreys' prior is the Hausdorff measure for the Hellinger and Kullback-Leibler distances (Q2714363)

Let \((\Omega, {\mathcal A}, F = \{ P_\vartheta\} _{\vartheta \in \Theta})\) be a dominated statistical experiment with \( \Theta \subset \mathbb R ^ k\) such that \( P_\vartheta \not= P_{\vartheta'} \) for \( \vartheta \neq \vartheta'\). Let \(\{ f_\vartheta \} _{\vartheta \in \Theta} \) be the corresponding family of \(\mu\)-densities. Let \( d(\vartheta, \vartheta ')\) and \( K(\vartheta, \vartheta ')\), respectively, denote the Hellinger distance and the Kullback-Leibler distance of \(P_\vartheta \) and \(P_{\vartheta'}\). Moreover assume that \( \vartheta \mapsto \sqrt{f_\vartheta} \in L^ 2 (\mu)\) is locally Lipschitz.NEWLINENEWLINE Using an area-formula for Hausdorff measures for (generalized) distances, the paper shows that the \(k\)-dimensional Hausdorff measure for \(d\) and the \(k/2\)-dimensional Hausdorff measure for \(K\) are both proportional to Jeffreys' prior.