IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i22p2845-d675833.html
   My bibliography  Save this article

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Author

Listed:
  • Sandra Fortini

    (Department of Decision Sciences, Bocconi University, 20136 Milano, Italy)

  • Sonia Petrone

    (Department of Decision Sciences, Bocconi University, 20136 Milano, Italy)

  • Hristo Sariev

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria)

Abstract

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k -color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP ( μ n ) n ≥ 0 on a Polish space X , the normalized sequence ( μ n / μ n ( X ) ) n ≥ 0 agrees with the marginal predictive distributions of some random process ( X n ) n ≥ 1 . Moreover, μ n = μ n − 1 + R X n , n ≥ 1 , where x ↦ R x is a random transition kernel on X ; thus, if μ n − 1 represents the contents of an urn, then X n denotes the color of the ball drawn with distribution μ n − 1 / μ n − 1 ( X ) and R X n —the subsequent reinforcement. In the case R X n = W n δ X n , for some non-negative random weights W 1 , W 2 , … , the process ( X n ) n ≥ 1 is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of ( X n ) n ≥ 1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.

Suggested Citation

  • Sandra Fortini & Sonia Petrone & Hristo Sariev, 2021. "Predictive Constructions Based on Measure-Valued Pólya Urn Processes," Mathematics, MDPI, vol. 9(22), pages 1-19, November.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2845-:d:675833
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/22/2845/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/9/22/2845/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Berti, Patrizia & Crimaldi, Irene & Pratelli, Luca & Rigo, Pietro, 2010. "Central limit theorems for multicolor urns with dominated colors," Stochastic Processes and their Applications, Elsevier, vol. 120(8), pages 1473-1491, August.
    2. Patrizia Berti & Irene Crimaldi & Luca Pratelli & Pietro Rigo, 2009. "Rate of Convergence of Predictive Distributions for Dependent Data," Quaderni di Dipartimento 091, University of Pavia, Department of Economics and Quantitative Methods.
    3. Sandra Fortini & Sonia Petrone, 2020. "Quasi‐Bayes properties of a procedure for sequential learning in mixture models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(4), pages 1087-1114, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Emanuele Dolera, 2022. "Preface to the Special Issue on “Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini’s 75th Birthday”," Mathematics, MDPI, vol. 10(15), pages 1-4, July.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Patrizia Berti & Luca Pratelli & Pietro Rigo, 2021. "A Central Limit Theorem for Predictive Distributions," Mathematics, MDPI, vol. 9(24), pages 1-11, December.
    2. Crimaldi, Irene & Louis, Pierre-Yves & Minelli, Ida G., 2022. "An urn model with random multiple drawing and random addition," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 270-299.
    3. Berti, Patrizia & Dreassi, Emanuela & Pratelli, Luca & Rigo, Pietro, 2021. "Asymptotics of certain conditionally identically distributed sequences," Statistics & Probability Letters, Elsevier, vol. 168(C).
    4. Crimaldi, Irene & Dai Pra, Paolo & Minelli, Ida Germana, 2016. "Fluctuation theorems for synchronization of interacting Pólya’s urns," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 930-947.
    5. Patrizia Berti & Luca Pratelli & Pietro Rigo, 2010. "Limit Theorems for Empirical Processes Based on Dependent Data," Quaderni di Dipartimento 132, University of Pavia, Department of Economics and Quantitative Methods.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2845-:d:675833. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.