Clustering on the Unit Hypersphere using von Mises-Fisher Distributions
The Journal of Machine Learning Research
The Journal of Machine Learning Research
Nonlinear Models Using Dirichlet Process Mixtures
The Journal of Machine Learning Research
The L1-consistency of Dirichlet mixtures in multivariate Bayesian density estimation
Journal of Multivariate Analysis
Statistics and Computing
Dirichlet Process Mixtures of Generalized Linear Models
The Journal of Machine Learning Research
Nonparametric Inference on Manifolds: With Applications to Shape Spaces
Nonparametric Inference on Manifolds: With Applications to Shape Spaces
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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.