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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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    Cited by:

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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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