Abstract
While identifiability of finite mixtures for a wide range of distributions has been studied by statisticians for decades, discussion on countably infinite mixtures is still limited. This article provides an sufficient condition by means of well-ordered sets and uniform convergence of series. It is then applied to revisit some examples for which the identifiability is well established and then explore the identifiability for several distribution families, including normal, gamma, Cauchy, noncentral \(\chi ^2\), multivariate normal distributions.
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References
Ahmad KE, Al-Hussaini EK (1982) Remarks on the non-identifiability of mixtures of distributions. Ann Inst Stat Math 34:543–544
Al-Hussaini EK, Ahmad KE (1981) On the identifiability of finite mixtures of distributions. IEEE Trans Inf Theory 27:664–668
Atienza N, Garcia-Heras J, Munoz-Pichardo JM (2006) A new condition for identifiability of finite mixture distributions. Metrika 63:215–221
Beal MJ, Krishnamurthy P (2012) Gene expression time course clustering with countably infinite hidden Markov models. arXiv, preprint arXiv:1206.6824
Bouguila N, Ziou D (2012) A countably infinite mixture model for clustering and feature selection. Knowl Inf Syst 33(2):351–370
Chandra S (1977) On the mixtures of probability distributions. Scand J Stat 3:105–112
DasGupta A (2008) Asymptotic theory of statistics and probability. Springer, New York
Doob JL (1949) Application of the theory of martingales. In: Le Calcul des probabilités et ses applications. Centre National de la Recherche Scientifique, Paris, pp 23–27
Escobar MD, West M (1995) Bayesian density estimation and inference using mixtures. J Am Stat Assoc 90(430):577–588
Escobar MD, West M (1998) Computing nonparametric hierarchical models. In: Practical nonparametric and semiparametric Bayesian statistics. Springer, New York, pp 1–22
Ferguson TS (1983) Bayesian density estimation by mixtures of normal distributions. In: Recent advances in statistics. Academic Press, New York, pp 287–302
Ghosal S, Ghosh JK, Ramamoorthi RV (1999) Posterior consistency of Dirichlet mixtures in density estimation. Ann Stat 27(1):143–158
Ghosal S, Ghosh JK, van der Vaart AW (2000) Convergence rates of posterior distributions. Ann Stat 28(2):500–531
Ghosal S, van der Vaart AW (2001) Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann Stat 29(5):1233–1263
Henna J (1994) Examples of identifiable mixture. J Jpn Stat Soc 24:193–200
Holzmann H, Munk A, Gneiting T (2006a) Identifiability of finite mixtures of elliptical distributions. Scand J Stat 33(4):753–763
Holzmann H, Munk A, Stratmann B (2004) Identifiability of finite mixtures - with application to circular distributions. Sankhyā 66(3):1–10
Holzmann H, Munk A, Zucchini W (2006b) On identifiability in capture-recapture models. Biometrics 62(3):934–936
Laird N (1978) Nonparametric maximum likelihood estimation of a mixing distribution. J Am Stat Assoc 73(364):805–811
Lindsay BG (1983a) The geometry of mixture likelihoods: a general theory. Ann Statist 11(1):86–94
Lindsay BG (1983b) The Geometry of mixture likelihoods, part II: the exponential family. Ann Stat 11(3):783–792
Lindsay BG (1995) Mixture models: theory, geometry and applications. IMS, Hayward
McLachlan G, Basford K (1988) Mixture models: inference and applications to clustering. Dekker, New York
Patil GP, Bildikar S (1966) Identifiability of countable mixtures of discrete probability distributions using methods of infinite matrices. Math Proc Camb Phil Soc 62(3):485–494
Reiersøl O (1950) Identifiability of a linear relation between variables which are subject to error. Econometrica 18:375–389
Shamilov A, Asma S (2009) On the identifiability of finite mixtures based on the union of logarithmic series and discrete rectangular families. Neural Process Lett 29:1–6
Tallis GM (1969) The identifiability of mixtures of distributions. J Appl Probab 6(2):389–398
Tallis GM, Chesson P (1982) Identifiability of mixtures. J Aust Math Soc 32(3):339–348
Teicher H (1961) Identifiability of mixtures. Ann Math Stat 32:244–248
Teicher H (1963) Identifiability of finite mixtures. Ann Math Stat 34:1265–1269
West M (1992) Modelling with mixtures. In: Bernaldo JM, Berger JO, Dawid AP, Smith AFM (eds) Bayesian statistics, vol 4. Oxford University Press, New York, pp 503–524
West M (1993) Approximating posterior distributions by mixtures. J R Stat Soc B 55(2):409–422
West M, Müller P, Escobar MD (1994) Hierarchical priors and mixture models, with application in regression and density estimation. In: Aspects of uncertainty. Wiley, Chichester, pp 363–386
Wu X, Zhou X (2006) A new characterization of distortion premiums via countable additivity for comonotonic risks. Insur Math Econ 38:324–334
Yakowitz SJ, Spragins JD (1968) On the identifiability of finite mixtures. Ann Math Stat 39:209–214
Acknowledgments
The authors are greatly grateful to an Associate Editor and a referee for their considerably helpful comments and suggestions, which eminently improve this essay. This research is partially supported by the Natural Science Foundation of China under Grant No. 71071056 and the Philosophy and Social Science Foundation of Shanghai under grant No. 2010BJB004.
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Yang, L., Wu, X. A new sufficient condition for identifiability of countably infinite mixtures. Metrika 77, 377–387 (2014). https://doi.org/10.1007/s00184-013-0444-x
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DOI: https://doi.org/10.1007/s00184-013-0444-x