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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
Software:
Octave
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References:
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