A Bayesian approach to the estimation of maps between riemannian manifolds
Abstract
Description
Let Θbe a smooth compact oriented manifold without boundary, embedded in a euclidean space and let γbe a smooth map Θinto a riemannian manifold Λ. An unknown state θ\in Θis observed via X=θ+εξwhere ε>0 is a small parameter and ξis a white Gaussian noise. For a given smooth prior on Θand smooth estimator g of the map γwe derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θand Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γis found based on the modern theory of harmonic maps and hypo-elliptic differential operators.
20 pages, no figures published version includes correction to eq.s 31, 41, 43
20 pages, no figures published version includes correction to eq.s 31, 41, 43