Model Misspecification, Bayesian Versus Credibility Estimation, and Gibbs Posteriors

22 Pages Posted: 18 Jul 2019 Last revised: 21 Nov 2019

See all articles by Liang Hong

Liang Hong

The University of Texas at Dallas

Ryan Martin

North Carolina State University - Department of Statistics

Date Written: July 18, 2019

Abstract

In the context of predicting future claims, a fully Bayesian analysis --- one that specifies a statistical model, prior distribution, and updates using Bayes's formula --- is often viewed as the gold-standard, while Buhlmann's credibility estimator serves as a simple approximation. But those desirable properties that give the Bayesian solution its elevated status depend critically on the posited model being correctly specified. Here we investigate the asymptotic behavior of Bayesian posterior distributions under a misspecified model, and our conclusion is that misspecification bias generally has damaging effects that can lead to inaccurate inference and prediction. The credibility estimator, on the other hand, is not sensitive at all to model misspecification, giving it an advantage over the Bayesian solution in those practically relevant cases where the model is uncertain. This begs the question: does robustness to model misspecification require that we abandon uncertainty quantification based on a posterior distribution? Our answer to this question is No, and we offer an alternative Gibbs posterior construction. Furthermore, we argue that this Gibbs perspective provides a new characterization of Buhlmann's credibility estimator.

Keywords: asymptotics; Bernstein--von Mises phenomenon; exponential family; robustness; uncertainty quantification

Suggested Citation

Hong, Liang and Martin, Ryan, Model Misspecification, Bayesian Versus Credibility Estimation, and Gibbs Posteriors (July 18, 2019). Available at SSRN: https://ssrn.com/abstract=3421780 or http://dx.doi.org/10.2139/ssrn.3421780

Liang Hong (Contact Author)

The University of Texas at Dallas ( email )

2601 North Floyd Road
Richardson, TX 75083
United States

Ryan Martin

North Carolina State University - Department of Statistics ( email )

Raleigh, NC 27695-8203
United States

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