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Calibrated Generalized Bayesian Inference

David T. Frazier, Christopher Drovandi, Robert Kohn

Abstract

We propose a simple approach that provides accurate uncertainty quantification for Bayesian inference in misspecified or approximate models, and for generalized (Gibbs) posteriors. While existing solutions in this context are based on explicit Gaussian approximations or post-processing procedures, we demonstrate that correct uncertainty quantification can be achieved by substituting the usual posterior with an intuitively appealing alternative that conveys the same information. This solution applies to both likelihood-based and loss-based posteriors, and is formally demonstrated to reliably quantify uncertainty. This new approach is demonstrated through a range of examples, including generalized linear models, and doubly intractable models.

Calibrated Generalized Bayesian Inference

Abstract

We propose a simple approach that provides accurate uncertainty quantification for Bayesian inference in misspecified or approximate models, and for generalized (Gibbs) posteriors. While existing solutions in this context are based on explicit Gaussian approximations or post-processing procedures, we demonstrate that correct uncertainty quantification can be achieved by substituting the usual posterior with an intuitively appealing alternative that conveys the same information. This solution applies to both likelihood-based and loss-based posteriors, and is formally demonstrated to reliably quantify uncertainty. This new approach is demonstrated through a range of examples, including generalized linear models, and doubly intractable models.
Paper Structure (34 sections, 8 theorems, 119 equations, 3 figures, 12 tables)

This paper contains 34 sections, 8 theorems, 119 equations, 3 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Bayesian Inference in Misspecified Models
  3. Setup and Known Issues
  4. Automatically Calibrated Bayesian Uncertainty
  5. Examples
  6. Linear Regression
  7. Poisson Regression
  8. Doubly Intractable Models
  9. DFD-Bayes: Conway-Maxwell-Poisson Model
  10. KSD-Bayes: Contaminated Normal
  11. Theoretical Results
  12. Assumptions
  13. Results
  14. Unique identification
  15. Non-unique Identification
  16. ...and 19 more sections

Key Result

Theorem 1

If Assumptions ass:infeasible-ass:prior hold and $\theta_\star$ is unique, then, for $p\in\{0,1\}$,

Figures (3)

  • Figure 1: Posterior summaries for data generated from a CMP model. Results are shown for ACP (green), the GB approach of matsubara2022robust (red), and an accurate approximation of the exact likelihood (blue).
  • Figure 2: Univariate posterior distributions for the mean parameter under Tukey's loss for ACP (green) and the Gaussian posterior correction (PostCorr) (red) when $\Theta_\star=\{-1,0,1\}$ (marked by crosses on the $x$-axis).
  • Figure 3: Univariate densities estimates of approximations to the posterior distribution for a single dataset generated from an ARFIMA with true parameter value $\theta = (\phi_1, \phi_2, \vartheta_1, d)^\top = (0.45, 0.1, -0.4, 0.4)^\top$. Shown are posterior approximations based on the Whittle likelihood (orange), ACP (green) and the exact likelihood (blue).

Theorems & Definitions (24)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 1
  • Corollary 1
  • Remark 8
  • ...and 14 more