Objective Bayesian inference with proper scoring rules

F. Giummolè (), V. Mameli (), E. Ruli () and L. Ventura ()
Additional contact information
F. Giummolè: Ca’ Foscari University of Venice
V. Mameli: Ca’ Foscari University of Venice
E. Ruli: University of Padova
L. Ventura: University of Padova

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2019, vol. 28, issue 3, No 9, 728-755

Abstract: 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 |

Keywords: $$\alpha $$ α -divergences; Composite likelihood; Godambe information; M-estimating function; Reference prior; Robustness; Scoring rule; 62F15; 62F35; 62E20 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s11749-018-0597-z

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