A Bayesian approach to the estimation of maps between riemannian manifolds

Details A Bayesian approach to the estimation of maps between riemannian manifolds A Bayesian approach to the estimation of maps between riemannian manifolds

2007-05-17

arxiv.org/abs/0705.2540v2

Mathematical Methods of Statistics. 16(4):1--17, 2007

Mathematics

Statistics Theory

20 pages, no figures published version includes correction to eq.s 31, 41, 43

Scientific paper

10.3103/S1066530707040011

Let \Theta be a smooth compact oriented manifold without boundary, embedded in a euclidean space and let \gamma be a smooth map \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 on \Theta and smooth estimator g 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 hypo-elliptic differential operators.

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