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Teor. Veroyatnost. i Primenen., 2000, Volume 45, Issue 1, Pages 103–124 (Mi tvp326)  

This article is cited in 6 scientific papers (total in 6 papers)

Approximation of laws of random probabilities by mixtures of Dirichlet distributions with applications to nonparametric Bayesian inference

E. Regazzinia, V. V. Sazonovb

a Dipartimento di Matematica, Università di Pavia, Itatia
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: In the general setting of nonparametric Bayesian inference, when observations are exchangeable and take values in a Polish space $X$, priors are approximated (in the Prokhorov metric) with any degree of precision by explicitly constructed mixtures of the distributions of Dirichlet processes. It is shown that if these mixtures ${\mathcal P}_{n}$ converge weakly to a given prior $\mathcal P$, the posteriors derived from ${\mathcal P}_{n}$'s converge weakly to the posterior deduced from $\mathcal P$. The error of approximation is estimated under some further assumptions. These results are applied to obtain a method for eliciting prior beliefs and to approximate both the predictive distribution (in the variational metric) and the posterior distribution function of $\int \psi d\widetilde{p}$ (in the Lévy metric), where $\widetilde p$ is a random probability having distribution $\mathcal P$.

Keywords: approximation of priors and posteriors, Dirichlet distributions, Dirichlet processes, elicitation of prior beliefs, Lévy metric, Prokhorov metric, random measures.

DOI: https://doi.org/10.4213/tvp326

Full text: PDF file (1037 kB)

English version:
Theory of Probability and its Applications, 2001, 45:1, 93–110

Bibliographic databases:

Received: 26.11.1998

Citation: E. Regazzini, V. V. Sazonov, “Approximation of laws of random probabilities by mixtures of Dirichlet distributions with applications to nonparametric Bayesian inference”, Teor. Veroyatnost. i Primenen., 45:1 (2000), 103–124; Theory Probab. Appl., 45:1 (2001), 93–110

Citation in format AMSBIB
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\jour Theory Probab. Appl.
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Regazzini E., Lijoi A., Prunster I., “Distributional results for means of normalized random measures with independent increments”, Annals of Statistics, 31:2 (2003), 560–585  crossref  mathscinet  zmath  isi  scopus
    2. Lijoi A., “Approximating priors by finite mixtures of conjugate distribution for an exponential family”, Journal of Statistical Planning and Inference, 113:2 (2003), 419–435  crossref  mathscinet  zmath  isi  scopus
    3. James L.F., Lijoi A., Prunster I., “On the posterior distribution of classes of random means”, Bernoulli, 16:1 (2010), 155–180  crossref  mathscinet  zmath  isi  elib  scopus
    4. Leisen F., Lijoi A., “Vectors of two-parameter Poisson-Dirichlet processes”, J Multivariate Anal, 102:3 (2011), 482–495  crossref  mathscinet  zmath  isi  elib  scopus
    5. Lijoi A., Pruenster I., “A Conversation with Eugenio Regazzini”, Stat. Sci., 26:4 (2011), 647–672  crossref  mathscinet  zmath  isi  scopus
    6. Cifarelli D.M., Dolera E., Regazzini E., “Frequentistic approximations to Bayesian prevision of exchangeable random elements”, Int. J. Approx. Reasoning, 78 (2016), 138–152  crossref  mathscinet  zmath  isi  elib  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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