A Bayesian approach to the estimation of maps between Riemannian manifolds

✍ Scribed by L. T. Butler; B. Levit

• Book ID: 111503687
• Publisher: Allerton Press Inc
• Year: 2007
• Tongue: English
• Weight: 799 KB
• Volume: 16
• Category: Article
• ISSN: 1066-5307
• DOI: 10.3103/s1066530707040011

✦ Synopsis

Let be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space , and let be a smooth map of into a Riemannian manifold Λ. An unknown state is observed via X = , where > 0 is a small parameter and is a white Gaussian noise. For a given smooth prior on and smooth estimators of the map we derive a secondorder asymptotic expansion for the related Bayesian risk. The calculation involves the geometry ofthe underlying spaces and , in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.