Journal
STATISTICA SINICA
Volume 27, Issue 2, Pages 535-553Publisher
STATISTICA SINICA
DOI: 10.5705/ss.202016.0017
Keywords
Bayesian nonparametric; kernel mixture; matrix Langevin; orthonormal matrices; posterior consistency; Stiefel manifold; von Mises Fisher
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Funding
- National Institute of Environmental Health Sciences (NIEHS) of the National Institute of Health (NIH) [R01ES017240]
- National Science Foundation (NSF) [IIS1546331]
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The Stiefel manifold V-p,V-d is the space of all d x p orthonormal matrices, with the d-1 hypersp here and the space of all orthogonal matrices constituting special cases. In modeling data lying on the Stiefel manifold, parametric distributions such as the matrix Langevin distribution are often used; however, model misspecification is a concern and it is desirable to have nonparametric alternatives. Current nonparametric methods are mainly Frechet-mean based. We take a fully generative nonparametric approach, which relies on mixing parametric kernels such as the matrix Langevin. The proposed kernel mixtures can approximate a large class of distributions on the Stiefel manifold, and we develop theory showing posterior consistency. While there exists work developing general posterior consistency results, extending these results to this particular manifold requires substantial new theory. Posterior inference is illustrated on a dataset of near-Earth objects.
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