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arXiv · 0705.2540

A Bayesian approach to the estimation of maps between riemannian manifolds

Leo T. Butler·Boris Levit
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Abstract

Let Θbe a smooth compact oriented manifold without boundary, embedded in a euclidean space and let γbe a smooth map Θinto a riemannian manifold Λ. An unknown state θ\in Θis observed via X=θ+εξwhere ε>0 is a small parameter and ξis a white Gaussian noise. For a given smooth prior on Θand smooth estimator g of the map γwe derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θand Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γis found based on the modern theory of harmonic maps and hypo-elliptic differential operators.

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Leo T. Butler, Boris Levit. 2008-03-25. A Bayesian approach to the estimation of maps between riemannian manifolds. https://doi.org/10.3103/s1066530707040011

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