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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.

##### MSC:
 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
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