Intrinsic predictive density estimation on the sphere under additional information

Abstract

We study predictive density estimation on the sphere under additional information. Specifically, we consider geodesic normal models on $$\mathbb {S}^p$$ in which the parameter pair $$(\mu _1,\mu _2)$$ is subject to an intrinsic restriction expressed through the logarithm map and geodesic distance. This yields a genuinely spherical analog of predictive density estimation with restricted parameters, without relying on ambient Euclidean differences. Under Kullback–Leibler loss, we derive exact Bayes predictive densities for both unrestricted and constrained product-uniform priors and show that the constrained rule is obtained from the unrestricted benchmark through a posterior-weight tilt induced by the restriction event. We further establish an orthogonally equivariant reduction of the predictive experiment and develop a local high-concentration expansion, showing that the spherical problem is asymptotically equivalent, at first order, to a Gaussian predictive problem with a restricted parameter difference in tangent coordinates, with curvature contributing only at the next order. These results yield exact Bayes-risk optimality under the restricted prior and local asymptotic dominance of the intrinsic constrained predictive density over the unrestricted benchmark. A numerical study on $$\mathbb {S}^1$$ illustrates the finite-concentration gains and the geometry-sensitive reshaping of the predictive density induced by the auxiliary information.