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Objective Bayesian inference with proper scoring rules

Author

Listed:
  • F. Giummolè

    (Ca’ Foscari University of Venice)

  • V. Mameli

    (Ca’ Foscari University of Venice)

  • E. Ruli

    (University of Padova)

  • L. Ventura

    (University of Padova)

Abstract

Standard Bayesian analyses can be difficult to perform when the full likelihood, and consequently the full posterior distribution, is too complex or even impossible to specify or if robustness with respect to data or to model misspecifications is required. In these situations, we suggest to resort to a posterior distribution for the parameter of interest based on proper scoring rules. Scoring rules are loss functions designed to measure the quality of a probability distribution for a random variable, given its observed value. Important examples are the Tsallis score and the Hyvärinen score, which allow us to deal with model misspecifications or with complex models. Also the full and the composite likelihoods are both special instances of scoring rules. The aim of this paper is twofold. Firstly, we discuss the use of scoring rules in the Bayes formula in order to compute a posterior distribution, named SR-posterior distribution, and we derive its asymptotic normality. Secondly, we propose a procedure for building default priors for the unknown parameter of interest that can be used to update the information provided by the scoring rule in the SR-posterior distribution. In particular, a reference prior is obtained by maximizing the average $$\alpha $$ α -divergence from the SR-posterior distribution. For $$0 \le |\alpha |

Suggested Citation

  • F. Giummolè & V. Mameli & E. Ruli & L. Ventura, 2019. "Objective Bayesian inference with proper scoring rules," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 728-755, September.
    • Handle: RePEc:spr:testjl:v:28:y:2019:i:3:d:10.1007_s11749-018-0597-z
    DOI: 10.1007/s11749-018-0597-z
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    References listed on IDEAS

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    1. Richard E. Chandler & Steven Bate, 2007. "Inference for clustered data using the independence loglikelihood," Biometrika, Biometrika Trust, vol. 94(1), pages 167-183.
    2. Alexander Dawid & Monica Musio, 2014. "Theory and applications of proper scoring rules," METRON, Springer;Sapienza Università di Roma, vol. 72(2), pages 169-183, August.
    3. Ghosh, J. K. & Mukerjee, Rahul, 1991. "Characterization of priors under which Bayesian and frequentist Bartlett corrections are equivalent in the multiparameter case," Journal of Multivariate Analysis, Elsevier, vol. 38(2), pages 385-393, August.
    4. Abhik Ghosh & Ayanendranath Basu, 2016. "Robust Bayes estimation using the density power divergence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 413-437, April.
    5. Nicole A. Lazar, 2003. "Bayesian empirical likelihood," Biometrika, Biometrika Trust, vol. 90(2), pages 319-326, June.
    6. A. Philip Dawid & Monica Musio & Laura Ventura, 2016. "Minimum Scoring Rule Inference," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 123-138, March.
    7. Malay Ghosh & Victor Mergel & Ruitao Liu, 2011. "A general divergence criterion for prior selection," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(1), pages 43-58, February.
    8. Walker, Stephen G., 2016. "Bayesian information in an experiment and the Fisher information distance," Statistics & Probability Letters, Elsevier, vol. 112(C), pages 5-9.
    9. In Hong Chang & Rahul Mukerjee, 2008. "Bayesian and frequentist confidence intervals arising from empirical-type likelihoods," Biometrika, Biometrika Trust, vol. 95(1), pages 139-147.
    10. Susanne M. Schennach, 2005. "Bayesian exponentially tilted empirical likelihood," Biometrika, Biometrika Trust, vol. 92(1), pages 31-46, March.
    11. J. Corcuera & F. Giummolè, 1998. "A Characterization of Monotone and Regular Divergences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(3), pages 433-450, September.
    12. A. Dawid, 2007. "The geometry of proper scoring rules," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(1), pages 77-93, March.
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    2. Arthur Pewsey & Eduardo García-Portugués, 2021. "Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 1-58, March.
    3. Jack Jewson & David Rossell, 2022. "General Bayesian loss function selection and the use of improper models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(5), pages 1640-1665, November.

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