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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

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

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

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

Bibliographic databases:

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
\Bibitem{RegSaz00} \by E.~Regazzini, V.~V.~Sazonov \paper Approximation of laws of random probabilities by mixtures of Dirichlet distributions with applications to nonparametric Bayesian inference \jour Teor. Veroyatnost. i Primenen. \yr 2000 \vol 45 \issue 1 \pages 103--124 \mathnet{http://mi.mathnet.ru/tvp326} \crossref{https://doi.org/10.4213/tvp326} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1810976} \zmath{https://zbmath.org/?q=an:0984.60030} \transl \jour Theory Probab. Appl. \yr 2001 \vol 45 \issue 1 \pages 93--110 \crossref{https://doi.org/10.1137/S0040585X97978063} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000167428900006} 

• http://mi.mathnet.ru/eng/tvp326
• https://doi.org/10.4213/tvp326
• http://mi.mathnet.ru/eng/tvp/v45/i1/p103

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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
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
3. James L.F., Lijoi A., Prunster I., “On the posterior distribution of classes of random means”, Bernoulli, 16:1 (2010), 155–180
4. Leisen F., Lijoi A., “Vectors of two-parameter Poisson-Dirichlet processes”, J Multivariate Anal, 102:3 (2011), 482–495
5. Lijoi A., Pruenster I., “A Conversation with Eugenio Regazzini”, Stat. Sci., 26:4 (2011), 647–672
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
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