Title: Objective Bayesian inference with proper scoring rules,
Abstract: Standard Bayesian analyses can be difficult to perform when the full
likelihood, and consequently the full posterior distribution, is too complex
and difficult 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 $\alpha-$divergence from the SR-posterior
distribution. For $0 \leq |\alpha|<1$, the result is a Jeffreys-type prior that
is proportional to the square root of the determinant of the Godambe
information matrix associated to the scoring rule. Some examples are discussed.