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A Bayesian approach to the estimation of maps between Riemannian manifolds. (English) Zbl 1229.62006
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.

62C10 Bayesian problems; characterization of Bayes procedures
62C20 Minimax procedures in statistical decision theory
62F12 Asymptotic properties of parametric estimators
53B20 Local Riemannian geometry
53C17 Sub-Riemannian geometry
Full Text: DOI arXiv
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