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Hierarchical reinforced urn processes

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  • Fortini, S.
  • Petrone, S.
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 82 (2012)
    Issue (Month): 8 ()
    Pages: 1521-1529

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    Handle: RePEc:eee:stapro:v:82:y:2012:i:8:p:1521-1529

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    Related research

    Keywords: Partial exchangeability; Mixtures of Markov chains; Hoppe’s urn; Infinite hidden Markov models; Bayesian nonparametrics;

    References

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    1. Pasquale Cirillo & Jürg Hüsler & Pietro Muliere, 2010. "A Nonparametric Urn-Based Approach To Interacting Failing Systems With An Application To Credit Risk Modeling," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(08), pages 1223-1240.
    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. 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.
    4. 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.
    5. 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.
    6. Lorenzo Trippa & Paolo Bulla & Sonia Petrone, 2011. "Extended Bernstein prior via reinforced urn processes," Annals of the Institute of Statistical Mathematics, Springer, vol. 63(3), pages 481-496, June.
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