Abstract
The Dirichlet-Ferguson measure is a random probability measure that has seen widespread use in Bayesian nonparametrics. Our main results can be seen as a first step towards the development of a stochastic analysis of the Dirichlet-Ferguson measure. We define a gradient that acts on functionals of the measure and derive its adjoint. The corresponding integration by parts formula is used to prove a covariance representation formula for square integrable functionals of the Dirichlet-Ferguson measure and to provide a quantitative central limit theorem for the first chaos. Our findings are illustrated by a variety of examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Blackwell, D., MacQueen, J.: Ferguson distribution via Pólya urn schemes. Ann. Stat. 1, 353–355 (1973)
Chen, L.H.Y., Goldstein, L., Shao, Q.: Normal Approximation by Stein’S Method. Springer, Berlin (2011)
Dello Schiavo, L., Lytvynov, E.: A Mecke-type characterization of the Dirichlet-Ferguson measure Preprint. ar**v:1706.07602v2 (2017)
Donnelly, P., Grimmett, G.: On the asymptotic distribution of large prime factors. J. Lond. Math. Soc. 47, 395–404 (1993)
Feng, S.: The Poisson-Dirichlet distribution and Related Topics. Springer, Berlin (2010)
Ferguson, T.S.: A Bayesian analysis of some non-parametric problems. Ann. Stat. 1, 209–230 (1973)
Fleming, V.H., Viot, M.: Some measure-valued Markov processes in population genetics theory. Indiana J. Math. 28, 817–843 (1979)
Gelman, A., Carlin, J., Stern, H., Dunson, D., Vehtari, A., Rubin, D.: Bayesian Data Analysis, 3rd edn. Chapman & Hall/CRC Texts in Statistical Science, London (2013)
Handa, K.: Quasi-invariant measures and their characterization by conditional probabilities. Bull. Sci. Math. 125, 583–604 (2001)
Last, G., Penrose, M.: Lectures on the Poisson Process. Cambridge University Press, Cambridge (2017)
Letac, G., Piccioni, M.: Dirichlet random walks. J. Appl. Probab. 51, 1081–1099 (2014)
Lijoi, A., Regazzini, E.: Means of a Dirichlet process and multiple hypergeometric functions. Ann. Probab. 32, 1469–1495 (2004)
Mecke, J.: Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. verw. Gebiete 9, 36–58 (1967)
Nguyen, T.D., Privault, N., Torrisi, G.L.: Gaussian estimates for the solutions of some one-dimensional stochastic equations. Potential Anal. 43, 289–311 (2015)
Nourdin, I., Peccati, G.: Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145, 75–118 (2009)
Nourdin, I., Peccati, G.: Normal approximations with Malliavin calculus. Cambridge University Press, Cambridge (2012)
Nualart, D., Vives, J.: Anticipative Calculus for the Poisson Process Based on the Fock Space Séminaire de Probabilités XXIV, pp 154–165. Springer-Verlag, New York (1988)
Peccati, G.: Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations. Ann. Probab. 32, 1796–1829 (2004)
Peccati, G.: Multiple integral representation for functionals of Dirichlet processes. Bernoulli 14, 91–124 (2008)
Peccati, G., Solé, J. L., Taqqu, M.S., Utzet, F.: Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38, 443–478 (2010)
Picard, J.: Formules de dualité sur l’espace de Poisson. Ann. de l’I H. P. Sect. B 32(4), 509–548 (1996)
Privault, N.: Chaotic and variational calculus in discrete and continuous time for the poisson process. Stochast. Stochast. Rep. 51(1–2), 83–109 (1994)
Privault, N.: Stochastic Analysis in Discrete and Continuous Settings. Springer, Berlin (2009)
Privault, N., Torrisi, G.L.: Probability approximation by Clark Ocone covariance representation. Electron. J. Probab. 18, 1–25 (2013)
Regazzini, E., Sazonov, V.V.: Approximation of laws of random probabilities by mixtures of Dirichlet distributions with applications to nonparametric Bayesian inference. Teor Veroyatnost. i Primenen. 45, 103–124 (2000)
Sethuraman, J.: A constructive definition of Dirichlet priors. Stat. Sin. 4, 639–650 (1994)
Teh, Y., Jordan, M., Beal, M., Blei, D.: Sharing Clusters among Related Groups: Hierarchical Dirichlet Processes. Advances in Neural Information Processing Systems 17 (NIPS 2004)
Tsilevich, N., Vershik, A., Yor, M.: Distinguished properties of the gamma process and related topics. ar**v:math/0005287 [math.PR] (2000)
Villani, C.: Optimal Transport. Old and New. Springer, Berlin (2009)
Zhang, X.: Clark-ocone formula and variational representation for poisson functionals. Ann. Probab. 32(4), 606–529 (2009)
Wu, L.: A new modified logarithmic Sobolev for Poisson point processes and several applications. Probab. Theory Relat. Fields 118, 427–438 (2000)
Acknowledgements
We thank the two anonymous referees whose insights have helped improve an earlier draft of the manuscript. IF was supported by Australian Research Council Grant No DP190100613. GLT would like to acknowledge support for the project titled “Epidemics and Counting Structures in the Erdos-Renyi Random Graphs” from the Istituto Nazionale di Alta Matematica “Francesco Severi”.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Flint, I., Torrisi, G.L. An Integration by Parts Formula for Functionals of the Dirichlet-Ferguson Measure, and Applications. Potential Anal 58, 703–730 (2023). https://doi.org/10.1007/s11118-021-09955-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-021-09955-8
Keywords
- Covariance representation
- Dirichlet process
- Gaussian approximation
- Integration by parts formula
- Stein’s method