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Consistency of mixture models with a prior on the number of components

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  • Miller Jeffrey W.

    (Department of Biostatistics, Harvard TH Chan School of Public Health, Boston, MA, United States)

Abstract

This article establishes general conditions for posterior consistency of Bayesian finite mixture models with a prior on the number of components. That is, we provide sufficient conditions under which the posterior concentrates on neighborhoods of the true parameter values when the data are generated from a finite mixture over the assumed family of component distributions. Specifically, we establish almost sure consistency for the number of components, the mixture weights, and the component parameters, up to a permutation of the component labels. The approach taken here is based on Doob’s theorem, which has the advantage of holding under extraordinarily general conditions, and the disadvantage of only guaranteeing consistency at a set of parameter values that has probability one under the prior. However, we show that in fact, for commonly used choices of prior, this yields consistency at Lebesgue-almost all parameter values, which is satisfactory for most practical purposes. We aim to formulate the results in a way that maximizes clarity, generality, and ease of use.

Suggested Citation

  • Miller Jeffrey W., 2023. "Consistency of mixture models with a prior on the number of components," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-9, January.
    • Handle: RePEc:vrs:demode:v:11:y:2023:i:1:p:9:n:1
    DOI: 10.1515/demo-2022-0150
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    References listed on IDEAS

    as
    1. Ishwaran H. & James L. F, 2001. "Gibbs Sampling Methods for Stick Breaking Priors," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 161-173, March.
    2. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265, October.
    3. Jeffrey W. Miller & Matthew T. Harrison, 2018. "Mixture Models With a Prior on the Number of Components," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(521), pages 340-356, January.
    4. Theofanis Sapatinas, 1995. "Identifiability of mixtures of power-series distributions and related characterizations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(3), pages 447-459, September.
    5. Weining Shen & Surya T. Tokdar & Subhashis Ghosal, 2013. "Adaptive Bayesian multivariate density estimation with Dirichlet mixtures," Biometrika, Biometrika Trust, vol. 100(3), pages 623-640.
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