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Urn schemes and reinforced random walks

Author

Listed:
  • Muliere, P.
  • Secchi, P.
  • Walker, S. G.

Abstract

We define a reinforced urn process (RUP) to be a reinforced random walk on a state space of urns and we show its partial exchangeability. When it is recurrent, a RUP is a mixture of Markov chains and we characterize its mixing distribution on the space of stochastic matrices. Many Bayesian nonparametric priors, like Pólya trees, the beta-Stacy process and, in general, neutral to the right processes can be derived from RUPs. Applications to survival data are examined.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:88:y:2000:i:1:p:59-78
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    Citations

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    Cited by:

    1. Cheng, Dan & Cirillo, Pasquale, 2018. "A reinforced urn process modeling of recovery rates and recovery times," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 1-17.
    2. Muliere, Pietro & Secchi, Piercesare & G. Walker, Stephen, 2003. "Reinforced random processes in continuous time," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 117-130, March.
    3. 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.
    4. Stefano Peluso & Antonietta Mira & Pietro Muliere & Alessandro Lomi, 2016. "International Trade: a Reinforced Urn Network Model," Papers 1601.03067, arXiv.org.
    5. Pasquale Cirillo & Jürg Hüsler & Pietro Muliere, 2013. "Alarm Systems and Catastrophes from a Diverse Point of View," Methodology and Computing in Applied Probability, Springer, vol. 15(4), pages 821-839, December.
    6. Pasquale Cirillo & Pietro Muliere, 2013. "An urn-based Bayesian block bootstrap," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(1), pages 93-106, January.
    7. 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.
    8. 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.
    9. John J. McCall, 2004. "Induction: From Kolmogorov and Solomonoff to De Finetti and Back to Kolmogorov," Metroeconomica, Wiley Blackwell, vol. 55(2‐3), pages 195-218, May.
    10. Paolo Bulla & Pietro Muliere, 2007. "Bayesian Nonparametric Estimation for Reinforced Markov Renewal Processes," Statistical Inference for Stochastic Processes, Springer, vol. 10(3), pages 283-303, October.
    11. 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.
    12. Andrea Arfè & Stefano Peluso & Pietro Muliere, 2021. "The semi-Markov beta-Stacy process: a Bayesian non-parametric prior for semi-Markov processes," Statistical Inference for Stochastic Processes, Springer, vol. 24(1), pages 1-15, April.
    13. Fortini, S. & Petrone, S., 2012. "Hierarchical reinforced urn processes," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1521-1529.
    14. Paolo Leonetti, 2018. "Finite Partially Exchangeable Laws Are Signed Mixtures of Product Laws," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 195-214, August.
    15. 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.
    16. Cirillo, Pasquale & Hüsler, Jürg, 2011. "Extreme shock models: An alternative perspective," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 25-30, January.
    17. Peluso, Stefano & Mira, Antonietta & Muliere, Pietro, 2015. "Reinforced urn processes for credit risk models," Journal of Econometrics, Elsevier, vol. 184(1), pages 1-12.

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