Non-parametric Bayesian inference via loss functions under model misspecification

TL;DR

This paper introduces a non-parametric Bayesian inference method based on loss functions, generalizing the Bayesian bootstrap with a Dirichlet process, which remains valid under model misspecification and is demonstrated through simulations.

Contribution

It develops a loss-based Bayesian non-parametric approach that extends the Bayesian bootstrap and ensures valid inference despite model misspecification.

Findings

Valid posterior distributions under misspecification

Consistent and asymptotically normal inference

Effective recovery of true parameters in simulations

Abstract

In the usual Bayesian setting, a full probabilistic model is required to link the data and parameters, and the form of this model and the inference and prediction mechanisms are specified via de Finetti's representation. In general, such a formulation is not robust to model misspecification of its component parts. An alternative approach is to draw inference based on loss functions, where the quantity of interest is defined as a minimizer of some expected loss, and to construct posterior distributions based on the loss-based formulation; this strategy underpins the construction of the Gibbs posterior. We develop a Bayesian non-parametric approach; specifically, we generalize the Bayesian bootstrap, and specify a Dirichlet process model for the distribution of the observables. We implement this using direct prior-to-posterior calculations, but also using predictive sampling. We also study the assessment of posterior validity for non-standard Bayesian calculations. We show that the developed non-standard Bayesian updating procedures yield valid posterior distributions in terms of consistency and asymptotic normality under model misspecification. Simulation studies show that the proposed methods can recover the true value of the parameter under misspecification.