# Objective Bayesian inference with proper scoring rules

arXiv: https://arxiv.org/abs/1711.10819 · HTML: https://mlsanity.com/paper/1711.10819/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Nov 2017.

## Built on by (most cited first, 12 of 12)

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- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks](https://mlsanity.com/paper/2112.08866/) (2021)
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- [Generalised Bayesian Inference for Discrete Intractable Likelihood](https://mlsanity.com/paper/2206.08420/) (2022)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks: An Extended Investigation](https://mlsanity.com/paper/2406.03154/) (2024)
- [Meta-Uncertainty in Bayesian Model Comparison](https://mlsanity.com/paper/2210.07278/) (2022)
- [Generalized Bayesian deep reinforcement learning](https://mlsanity.com/paper/2412.11743/) (2024)
- [SMRS: advocating a unified reporting standard for surrogate models in the artificial intelligence era](https://mlsanity.com/paper/2502.06753/) (2025)
- [Differentially Private Statistical Inference through β-Divergence One Posterior Sampling](https://mlsanity.com/paper/2307.05194/) (2023)
- [Sparse Bayesian Correntropy Learning for Robust Muscle Activity Reconstruction from Noisy Brain Recordings](https://mlsanity.com/paper/2404.15309/) (2024)
- [Locking and Quacking: Stacking Bayesian model predictions by log-pooling and superposition](https://mlsanity.com/paper/2305.07334/) (2023)
- [Correcting Mode Proportion Bias in Generalized Bayesian Inference via a Weighted Kernel Stein Discrepancy](https://mlsanity.com/paper/2503.02108/) (2025)

## Abstract (arXiv)

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 α-divergence from the SR-posterior distribution. For 0 ≤ |α|<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.