## The Annals of Statistics

### Functionals of Dirichlet processes, the Cifarelli–Regazzini identity and Beta-Gamma processes

Lancelot F. James

#### Abstract

Suppose that Pθ(g) is a linear functional of a Dirichlet process with shape θH, where θ>0 is the total mass and H is a fixed probability measure. This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and a posterior calculus for Gamma processes to ascertain properties of linear functionals of Dirichlet processes. In particular, in conjunction with a Gamma identity, we show easily that a generalized Cauchy–Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, Beta-Gamma processes. This represents a generalization of an identity due to Cifarelli and Regazzini, which is also known as the Markov–Krein identity for mean functionals of Dirichlet processes. These results also provide new explanations and interpretations of results in the literature. The identities are analogues to quite useful identities for Beta and Gamma random variables. We give a result which can be used to ascertain specifications on H such that the Dirichlet functional is Beta distributed. This avoids the need for an inversion formula for these cases and points to the special nature of the Dirichlet process, and indeed the functional Beta-Gamma calculus developed in this paper.

#### Article information

Source
Ann. Statist., Volume 33, Number 2 (2005), 647-660.

Dates
First available in Project Euclid: 26 May 2005

https://projecteuclid.org/euclid.aos/1117114332

Digital Object Identifier
doi:10.1214/009053604000001237

Mathematical Reviews number (MathSciNet)
MR2163155

Zentralblatt MATH identifier
1071.62026

Subjects
Primary: 62G05: Estimation
Secondary: 62F15: Bayesian inference

#### Citation

James, Lancelot F. Functionals of Dirichlet processes, the Cifarelli–Regazzini identity and Beta-Gamma processes. Ann. Statist. 33 (2005), no. 2, 647--660. doi:10.1214/009053604000001237. https://projecteuclid.org/euclid.aos/1117114332

#### References

• Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2 1152--1174.
• Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 353--355.
• Cifarelli, D. M. and Melilli, E. (2000). Some new results for Dirichlet priors. Ann. Statist. 28 1390--1413.
• Cifarelli, D. M. and Regazzini, E. (1990). Distribution functions of means of a Dirichlet process. Ann. Statist. 18 429--442.
• Diaconis, P. and Freedman, D. A. (1999). Iterated random functions. SIAM Rev. 41 45--76.
• Diaconis, P. and Kemperman, J. (1996). Some new tools for Dirichlet priors. In Bayesian Statistics 5 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 97--106. Oxford Univ. Press.
• Doss, H. and Sellke, T. (1982). The tails of probabilities chosen from a Dirichlet prior. Ann. Statist. 10 1302--1305.
• Dufresne, D. (1998). Algebraic properties of beta and gamma distributions, and applications. Adv. in Appl. Math. 20 285--299.
• Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 87--112.
• Feigin, P. and Tweedie, R. (1989). Linear functionals and Markov chains associated with the Dirichlet processes. Math. Proc. Cambridge Philos. Soc. 105 579--585.
• Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 209--230.
• Freedman, D. A. (1963). On the asymptotic behavior of Bayes estimates in the discrete case. Ann. Math. Statist. 34 1386--1403.
• Gupta, R. D. and Richards, D. St. P. (2001). The history of the Dirichlet and Liouville distributions. Internat. Statist. Rev. 69 433--446.
• Hjort, N. L. (2003). Topics in non-parametric Bayesian statistics. In Highly Structured Stochastic Systems (P. J. Green, N. L. Hjort and S. Richardson, eds.) 455--487. Oxford Univ. Press.
• James, L. F. (2002). Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics. Available at http://arxiv.org/abs/math.PR/0205093.
• Kerov, S. (1998). Interlacing measures. Amer. Math. Soc. Transl. Ser. 2 181 35--83.
• Kerov, S. and Tsilevich, N. V. (1998). The Markov--Krein correspondence in several dimensions. PDMI Preprint 1/1998, Steklov Institute of Mathematics at St. Petersburg.
• Lijoi, A. and Regazzini, E. (2004). Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32 1469--1495.
• Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates. I. Density estimates Ann. Statist. 12 351--357.
• Lo, A. Y. and Weng, C.-S. (1989). On a class of Bayesian nonparametric estimates. II. Hazard rate estimates. Ann. Inst. Statist. Math. 41 227--245.
• Pitman, J. (1996). Some developments of the Blackwell--MacQueen urn scheme. In Statistics, Probability and Game Theory (T. S. Ferguson, L. S. Shapley and J. B. MacQueen, eds.) 245--267. IMS, Hayward, CA.
• Regazzini, E., Guglielmi, A. and Di Nunno, G. (2002). Theory and numerical analysis for exact distributions of functionals of a Dirichlet process. Ann. Statist. 30 1376--1411.
• Regazzini, E., Lijoi, A. and Prünster, I. (2003). Distributional results for means of normalized random measures with independent increments. Ann. Statist. 31 560--585.
• Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statist. Sinica 4 639--650.
• Vershik, A. M., Yor, M. and Tsilevich, N. V. (2001). Remarks on the Markov--Krein identity and quasi-invariance of the gamma process. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. $($POMI$)$ 283 21--36. [In Russian. Translation in J. Math. Sci. (N. Y.) 121 (2004) 2303--2310.]