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Nonparametric Bayes classification and hypothesis testing on manifolds

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  • Bhattacharya, Abhishek
  • Dunson, David

Abstract

Our first focus is prediction of a categorical response variable using features that lie on a general manifold. For example, the manifold may correspond to the surface of a hypersphere. We propose a general kernel mixture model for the joint distribution of the response and predictors, with the kernel expressed in product form and dependence induced through the unknown mixing measure. We provide simple sufficient conditions for large support and weak and strong posterior consistency in estimating both the joint distribution of the response and predictors and the conditional distribution of the response. Focusing on a Dirichlet process prior for the mixing measure, these conditions hold using von Mises–Fisher kernels when the manifold is the unit hypersphere. In this case, Bayesian methods are developed for efficient posterior computation using slice sampling. Next we develop Bayesian nonparametric methods for testing whether there is a difference in distributions between groups of observations on the manifold having unknown densities. We prove consistency of the Bayes factor and develop efficient computational methods for its calculation. The proposed classification and testing methods are evaluated using simulation examples and applied to spherical data applications.

Suggested Citation

  • Bhattacharya, Abhishek & Dunson, David, 2012. "Nonparametric Bayes classification and hypothesis testing on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 1-19.
  • Handle: RePEc:eee:jmvana:v:111:y:2012:i:c:p:1-19
    DOI: 10.1016/j.jmva.2012.02.020
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    References listed on IDEAS

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    1. Wu, Yuefeng & Ghosal, Subhashis, 2010. "The L1-consistency of Dirichlet mixtures in multivariate Bayesian density estimation," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2411-2419, November.
    2. Basu S. & Chib S., 2003. "Marginal Likelihood and Bayes Factors for Dirichlet Process Mixture Models," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 224-235, January.
    3. Abhishek Bhattacharya & David B. Dunson, 2010. "Nonparametric Bayesian density estimation on manifolds with applications to planar shapes," Biometrika, Biometrika Trust, vol. 97(4), pages 851-865.
    4. Bigelow, Jamie L. & Dunson, David B., 2009. "Bayesian Semiparametric Joint Models for Functional Predictors," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 26-36.
    5. Michael L. Pennell & David B. Dunson, 2008. "Nonparametric Bayes Testing of Changes in a Response Distribution with an Ordinal Predictor," Biometrics, The International Biometric Society, vol. 64(2), pages 413-423, June.
    6. David B. Dunson & Shyamal D. Peddada, 2008. "Bayesian nonparametric inference on stochastic ordering," Biometrika, Biometrika Trust, vol. 95(4), pages 859-874.
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    Cited by:

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    5. You, Kisung & Suh, Changhee, 2022. "Parameter estimation and model-based clustering with spherical normal distribution on the unit hypersphere," Computational Statistics & Data Analysis, Elsevier, vol. 171(C).

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