zbMATH — the first resource for mathematics

Posterior consistency of Dirichlet mixtures in density estimation. (English) Zbl 0932.62043
Summary: A Dirichlet mixture of normal densities is a useful choice for a prior distribution on densities in the problem of Bayesian density estimation. In the recent years, an efficient Markov chain Monte Carlo method for the computation of the posterior distribution has been developed. The method has been applied to data arising from different fields of interest. The important issue of consistency was however left open. In this paper, we settle this issue in affirmative.

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62F15 Bayesian inference
Full Text: DOI
[1] Barron, A. R. (1988). The exponential convergence of posterior probabilities with implications for Bayes estimators of density functions. Unpublished manuscript.
[2] Barron, A. R. (1989). Uniformly powerful goodness of fit tests. Ann. Statist. 17 107-124. · Zbl 0674.62032
[3] Barron, A. R. (1998). Information theoretic analysis of Bayes performance and the choice of priors in parametric and nonparametric problems. In Bayesian Statistics 6 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) Oxford Univ. Press. · Zbl 0974.62020
[4] Barron, A. R., Schervish, M. and Wasserman, L. (1998). The consistency of posterior distributions in nonparametric problems. Ann. Statist. · Zbl 0980.62039
[5] Doss, H. and Sellke, T. (1982). The tails of probabilities chosen from a Dirichlet prior. Ann. Statist. 10 1302-1305. · Zbl 0515.62008
[6] Ferguson, T. S. (1983). Bayesian density estimation by mixtures of Normal distributions. In Recent Advances in Statistics (M. Rizvi, J. Rustagi, and D. Siegmund, eds.) 287-302. Academic Press, New York. · Zbl 0557.62030
[7] Gasparini, M. (1992). Bayes nonparametrics for biased sampling and density estimation. Ph.D. dissertation, Univ. Michigan.
[8] Ghorai, J. K. and Rubin, H. (1982). Bayes risk consistency of nonparametric Bayes density estimates. Austral. J. Statist. 24 51-66. · Zbl 0486.62041
[9] Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates I: density estimates. Ann. Statist. 12 351-357. · Zbl 0557.62036
[10] LeCam, L. (1973). Convergence of estimates under dimensionality restrictions. Ann. Statist. 1 38-53. · Zbl 0255.62006
[11] Roeder K. and Wasserman, L. (1995). Practical Bayesian density estimation using mixtures of normal. J. Amer. Statist. Assoc. 92 894-902. JSTOR: · Zbl 0889.62021
[12] Schwartz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10-26. · Zbl 0158.17606
[13] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[14] West, M. (1992). Modeling with mixtures. In Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds.) 503-524. Oxford Univ. Press.
[15] West, M., Muller, P. and Escobar, M. D. (1994). Hierarchical priors and mixture models, with applications in regression and density estimation. In Aspects of Uncertainty: A Tribute to D. V. Lindley (P. R. Freeman and A. F. M. Smith, eds.) 363-386. Wiley, Chichester. · Zbl 0842.62001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.