an:05277713
Zbl 1229.62006
Butler, L. T.; Levit, B.
A Bayesian approach to the estimation of maps between Riemannian manifolds
EN
Math. Methods Stat. 16, No. 4, 281-297 (2007).
00216016
2007
j
62C10 62C20 62F12 53B20 53C17
Bayes estimators; minimax estimators; sub-Riemannian geometry; sub-Laplacian; harmonic maps
Summary: Let \(\Theta \) be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space \(E^{s}\), and let \(\gamma \) be a smooth map of \(\Theta \) into a Riemannian manifold \(\Lambda \). An unknown state \(\theta \in \Theta \) is observed via \(X = \theta + \varepsilon \xi \), where \(\varepsilon > 0\) is a small parameter and \(\xi \) is a white Gaussian noise. For a given smooth prior \(\lambda \) on \(\Theta \) and smooth estimators \(g(X)\) of the map \(\gamma \) we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces \(\Theta \) and \(\Lambda \), in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of \(\gamma \) is found based on the modern theory of harmonic maps and hypoelliptic differential operators.