A Bayesian approach to the estimation of maps between Riemannian manifolds. II: Examples

Abstract

Let gJ be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space E s, and let γ be a smooth map of gJ into a Riemannian manifold Λ. An unknown state θ ∈ gJ is observed via X = θ + ɛξ, where ɛ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on gJ and smooth estimators g(X) of the map γ we have derived a second-order asymptotic expansion for the related Bayesian risk [3]. In this paper, we apply this technique to a variety of examples.The second part examines the first-order conditions for equality-constrained regression problems. The geometric tools that are utilized in [3] are naturally applicable to these regression problems.