Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-29T19:23:02.882Z Has data issue: false hasContentIssue false
  • English
  • Français

Sufficientness postulates for measure-valued Pólya urn sequences

Part of: Sufficiency and information Nonparametric inference Stochastic processes

Published online by Cambridge University Press:  28 March 2025

Hristo Sariev Open the ORCID record for Hristo Sariev [Opens in a new window]  and
Mladen Savov
Hristo Sariev*
Affiliation:
Bulgarian Academy of Sciences
Mladen Savov*
Affiliation:
Sofia University ‘St. Kliment Ohridski’
*
*Postal address: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad. Georgi Bonchev Str., Sofia 1113, Bulgaria. Email: h.sariev@math.bas.bg
**Postal address: Faculty of Mathematics and Informatics, Sofia University “St. Kliment Ohridski”, 5 James Bourchier Blvd, Sofia 1164, Bulgaria. Email: msavov@fmi.uni-sofia.bg
Get access Rights & Permissions [Opens in a new window]

Abstract

In a recent paper, the authors studied the distribution properties of a class of exchangeable processes, called measure-valued Pólya sequences (MVPSs), which arise as the observation process in a generalized urn sampling scheme. Here we present several results in the form of ‘sufficientness’ postulates that characterize their predictive distributions. In particular, we show that exchangeable MVPSs are the unique exchangeable models whose predictive distributions are a mixture of the marginal distribution and the average of a probability kernel evaluated at past observations. When the latter coincides with the empirical measure, we recover a well-known result for the exchangeable model with a Dirichlet process prior. In addition, we provide a ‘pure’ sufficientness postulate for exchangeable MVPSs that does not assume a particular analytic form for the predictive distributions. Two other sufficientness postulates consider the case when the state space is finite.

Keywords

Pólya urnspredictive distributionsBayesian nonparametricsexchangeabilitysufficientness postulate

MSC classification

Primary: 60G25: Prediction theory 60G09: Exchangeability
Secondary: 62G99: None of the above, but in this section 62B99: None of the above, but in this section

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 3 , September 2025 , pp. 925 - 949
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aldous, D. J. (1985). Exchangeability and related topics. In École d’Été de Probabilités de Saint-Flour XIII – 1983 (Lect. Notes Math. 1117), eds Aldous, D. J., Ibragimov, I. A. and Jacod, J.. Springer, Berlin, pp. 1–198.CrossRefGoogle Scholar
Bacallado, S., Battiston, M., Favaro, S. and Trippa, L. (2017). Sufficientness postulates for Gibbs-type priors and hierarchical generalizations. Statist. Sci. 32, 487–500.10.1214/17-STS619CrossRefGoogle Scholar
Bandyopadhyay, A. and Thacker, D. (2022). A new approach to Pólya urn schemes and its infinite color generalization. Ann. Appl. Prob. 32, 46–79.10.1214/21-AAP1671CrossRefGoogle Scholar
Berti, P., Dreassi, E., Leisen, F., Pratelli, L. and Rigo, P. (2023). Kernel-based Dirichlet sequences. Bernoulli 29, 1321–1342.10.3150/22-BEJ1500CrossRefGoogle Scholar
Berti, P., Dreassi, E., Leisen, F., Pratelli, L. and Rigo, P. (2023). A probabilistic view on predictive constructions for Bayesian learning. To appear in Statist. Sci.Google Scholar
Berti, P. and Rigo, P. (2007). 0–1 laws for regular conditional distributions. Ann. Prob. 35, 649–662.10.1214/009117906000000845CrossRefGoogle Scholar
Blackwell, D. and MacQueen, J. B. (1973). Ferguson distributions via Pólya urn schemes. Ann. Statist. 1, 353–355.10.1214/aos/1176342372CrossRefGoogle Scholar
Camerlenghi, F. and Favaro, S. (2021). On Johnson’s ‘sufficientness’ postulates for feature-sampling models. Mathematics 9, 2891.10.3390/math9222891CrossRefGoogle Scholar
Fong, E., Holmes, C. and Walker, S. G. (2023). Martingale posterior distributions. J. R. Statist. Soc. B 85, 1357–1391.10.1093/jrsssb/qkad005CrossRefGoogle Scholar
Fortini, S., Ladelli, L. and Regazzini, E. (2000). Exchangeability, predictive distributions and parametric models. Sankhyā A 62, 86–109.Google Scholar
Fortini, S. and Petrone, S. (2023). Prediction-based uncertainty quantification for exchangeable sequences. Phil. Trans. R. Soc. A 381, 20220142.10.1098/rsta.2022.0142CrossRefGoogle ScholarPubMed
Fortini, S. and Petrone, S. (2024). Exchangeability, prediction and predictive modeling in Bayesian statistics. Statist. Sci. 40, 40–67.Google Scholar
Fortini, S., Petrone, S. and Sariev, H. (2021). Predictive constructions based on measure-valued Pólya urn processes. Mathematics 9, 2845.10.3390/math9222845CrossRefGoogle Scholar
Garibaldi, U. and Scalas, E. (2010). Finitary Probabilistic Methods in Econophysics. Cambridge University Press.10.1017/CBO9780511777585CrossRefGoogle Scholar
Ghosal, S. and van der Vaart, A. (2017). Fundamentals of Nonparametric Bayesian Inference. Cambridge University Press.10.1017/9781139029834CrossRefGoogle Scholar
Hansen, B. and Pitman, J. (2000). Prediction rules for exchangeable sequences related to species sampling. Statist. Prob. Lett. 46, 251–256.10.1016/S0167-7152(99)00109-1CrossRefGoogle Scholar
Hill, B. M., Lane, D. and Sudderth, W. (1987). Exchangeable urn processes. Ann. Prob. 15, 1586–1592.10.1214/aop/1176991995CrossRefGoogle Scholar
Janson, S. (2019). Random replacements in Pólya urns with infinitely many colours. Electron. Commun. Prob. 24, 23.10.1214/19-ECP226CrossRefGoogle Scholar
Johnson, W. E. (1932). Probability: The deductive and inductive problems. Minds 41, 409–423.Google Scholar
Kallenberg, O. (2021). Foundations of Modern Probability, 3rd edn. Springer, New York.10.1007/978-3-030-61871-1CrossRefGoogle Scholar
Lange, K. (1973). Decompositions of substochastic transition functions. Proc. Amer. Math. Soc. 37, 575–580.10.1090/S0002-9939-1973-0314124-4CrossRefGoogle Scholar
Lijoi, A. and Prünster, I. (2010). Models beyond the Dirichlet process. In Bayesian Nonparametrics, eds Hjort, N. L., Holmes, C., Müller, P., and Walker, S. G.. Cambridge University Press, pp. 80–136.10.1017/CBO9780511802478.004CrossRefGoogle Scholar
Lo, A. Y. (1991). A characterization of the Dirichlet process. Statist. Prob. Lett. 13, 185–187.10.1016/0167-7152(91)90075-3CrossRefGoogle Scholar
Mailler, C. and Marckert, J.-F. (2017). Measure-valued Pólya urn processes. Electron. J. Prob. 22, 26.10.1214/17-EJP47CrossRefGoogle Scholar
Pitman, J. (1996). Some developments of the Blackwell–MacQueen urn scheme. In Statistics, Probability and Game Theory: Papers in Honor of David Blackwell (IMS Lect. Notes Monogr. Ser. 30), eds Ferguson, T. S., Shapley, L. S. and MacQueen, J. B.. Institute of Mathematical Statistics, 245–267.10.1214/lnms/1215453576CrossRefGoogle Scholar
Regazzini, E. (1978). Intorno ad alcune questioni relative alla definizione del premio secondo la teoria della credibilità. G. Ist. Ital. Attuari 41, 77–89.Google Scholar
Sariev, H. and Savov, M. (2024). Characterization of exchangeable measure-valued Pólya urn sequences. Electron. J. Prob. 29, 73.10.1214/24-EJP1132CrossRefGoogle Scholar
Walker, S. G. and Muliere, P. (1999). A characterization of a neutral to the right prior via an extension of Johnson’s sufficientness postulate. Ann. Statist. 27, 589–599.Google Scholar
Zabell, S. L. (1982). W. E. Johnson’s ‘sufficientness’ postulate. Ann. Statist. 10, 1090–1099.10.1214/aos/1176345975CrossRefGoogle Scholar
Zabell, S. L. (1995). Characterizing Markov exchangeable sequences. J. Theoret. Prob. 8, 175–178.10.1007/BF02213460CrossRefGoogle Scholar
Zabell, S. L. (1997). The continuum of inductive methods revisited. In The Cosmos of Science: Essays in Exploration, eds. Earman, J. and Norton, J. D.. University of Pittsburgh Press, pp. 351–385.Google Scholar