A Bayesian Approach to the Estimation of Maps between Riemannian Manifolds
نویسندگان
چکیده
Let Θ be a smooth compact oriented manifold without boundary, imbedded in a euclidean space E, and let γ be a smooth map of Θ into a Riemannian manifold Λ. An unknown state θ ∈ Θ is observed via X = θ + ǫξ where ǫ > 0 is a small parameter and ξ is a white Gaussian noise. For a given smooth prior λ on Θ and smooth estimators g(X) of the map γ we derive a second-order asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of the underlying spaces Θ and Λ, in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of γ is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.
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