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A Bayesian approach to the estimation of maps between Riemannian manifolds. II: examples. (English) Zbl 1282.62017
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 $$X=\theta+\epsilon\xi$$, where $$\epsilon>0$$ 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 have derived a second-order asymptotic expansion for the related Bayesian risk [the authors, ibid. 16, No. 4, 281–297 (2007; Zbl 1229.62006)]. 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 [the authors, loc. cit.] are naturally applicable to these regression problems.
##### 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
Octave
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##### References:
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