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Hierarchical reinforced urn processes

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  • Fortini, S.
  • Petrone, S.

Abstract

We define a class of reinforced urn processes, based on Hoppe’s urn scheme, that are Markov exchangeable, with a countable and possibly unknown state space. This construction extends the reinforced urn processes developed by Muliere et al. (2000) and widely used in Bayesian nonparametric inference and survival analysis. We also shed light on the connections with apparently unrelated constructions, recently proposed in the machine learning literature, such as the infinite hidden Markov model, offering a general framework for a deeper study of their theoretical properties.

Suggested Citation

  • Fortini, S. & Petrone, S., 2012. "Hierarchical reinforced urn processes," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1521-1529.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:8:p:1521-1529
    DOI: 10.1016/j.spl.2012.04.012
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    References listed on IDEAS

    as
    1. Fortini, Sandra & Ladelli, Lucia & Petris, Giovanni & Regazzini, Eugenio, 0. "On mixtures of distributions of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 147-165, July.
    2. Muliere, P. & Secchi, P. & Walker, S. G., 2000. "Urn schemes and reinforced random walks," Stochastic Processes and their Applications, Elsevier, vol. 88(1), pages 59-78, July.
    3. Lorenzo Trippa & Paolo Bulla & Sonia Petrone, 2011. "Extended Bernstein prior via reinforced urn processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(3), pages 481-496, June.
    4. Teh, Yee Whye & Jordan, Michael I. & Beal, Matthew J. & Blei, David M., 2006. "Hierarchical Dirichlet Processes," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1566-1581, December.
    5. Muliere, Pietro & Secchi, Piercesare & Walker, Stephen, 2005. "Partially exchangeable processes indexed by the vertices of a k-tree constructed via reinforcement," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 661-677, April.
    6. Mezzetti, Maura & Muliere, Pietro & Bulla, Paolo, 2007. "An application of reinforced urn processes to determining maximum tolerated dose," Statistics & Probability Letters, Elsevier, vol. 77(7), pages 740-747, April.
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    Cited by:

    1. Dan Cheng & Pasquale Cirillo, 2019. "An Urn-Based Nonparametric Modeling of the Dependence between PD and LGD with an Application to Mortgages," Risks, MDPI, vol. 7(3), pages 1-21, July.
    2. Souto Arias, Luis A. & Cirillo, Pasquale, 2021. "Joint and survivor annuity valuation with a bivariate reinforced urn process," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 174-189.

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