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Activity Number: 553 - Bayesian Nonparametrics
Type: Invited
Date/Time: Wednesday, August 2, 2017 : 2:00 PM to 3:50 PM
Sponsor: IMS
Abstract #322184
Title: Singularity Structure of Parameter Space and Posterior Contraction in Finite Mixture Models
Author(s): Long Nguyen and Nhat Ho*
Companies: University of Michigan and University of Michigan
Keywords: singularities ; Fisher information ; mixing distribution ; Wasserstein distance ; posterior contraction ; convergence rates
Abstract:

Singularities of a statistical model are the elements of the model's parameter space which make the corresponding Fisher information matrix degenerate. These are the points for which estimation techniques such as the maximum likelihood estimator and standard Bayesian procedures do not admit the root-n parametric rate of convergence. We propose a general framework for the identification of singularity structures of the parameter space of finite mixtures, and study the impacts of the singularity levels on parameter estimation in a compact parameter space. Our study makes explicit the links between model singularities, parameter estimation convergence rates, and the algebraic geometry of the parameter space for mixtures of continuous distributions. The theory is applied to establish concrete convergence rates of both Bayesian and maximum likelihood parameter estimation for finite mixtures of skewnormal distributions. This rich and increasingly popular mixture model is shown to exhibit a remarkably complex range of asymptotic behaviors which have not been hitherto reported in the literature.


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