Objective Bayesian inference with proper scoring rules

TL;DR

This paper introduces a Bayesian inference approach using proper scoring rules like Tsallis and Hyv"arinen scores, enabling robust analysis with complex or misspecified models, and proposes default priors based on information divergence.

Contribution

It develops a framework for Bayesian inference with scoring rules, derives asymptotic properties of the SR-posterior, and constructs default priors using divergence maximization.

Findings

SR-posterior distribution is asymptotically normal.

Default priors are proportional to the square root of the Godambe information.

Examples illustrate the application of the proposed methods.

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 Hyv\"arinen 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.