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Bayesian inference for generalized linear models via quasi-posteriors.

D Agnoletto1, T Rigon2, D B Dunson1

  • 1Department of Statistical Science, Duke University, 214 Old Chemistry, Durham, North Carolina 27708, USA.

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Summary
This summary is machine-generated.

This study introduces quasi-posterior distributions for robust Bayesian inference in generalized linear models. This method enhances reliability by requiring only the first two moments to be correctly specified, improving upon traditional models.

Keywords:
C-BayesGeneralized BayesModel misspecificationQuasilikelihoodRobustness

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Area of Science:

  • Statistics
  • Bayesian Inference
  • Generalized Linear Models

Background:

  • Generalized linear models (GLMs) are widely used but susceptible to model misspecification, potentially affecting inferential accuracy.
  • Frequentist approaches like quasilikelihood offer robustness by relying on moment conditions, but a parallel Bayesian method was needed.
  • Existing Bayesian methods may lack robustness when model assumptions are violated.

Purpose of the Study:

  • To develop a robust Bayesian inference framework for generalized linear models using quasi-posterior distributions.
  • To establish a coherent generalized Bayes inference method that is robust to model misspecification.
  • To provide new insights into the selection of coarsening parameters and their interpretation.

Main Methods:

  • Development of quasi-posterior distributions as a novel approach to Bayesian inference in GLMs.
  • Theoretical analysis of the asymptotic properties of quasi-posteriors, including convergence and connections to other methods.
  • Investigation of the interpretation and application of the loss-scale parameter as a measure of dispersion.

Main Results:

  • Quasi-posterior distributions offer a coherent generalized Bayes inference method, approximating coarsened posteriors.
  • Asymptotically, the quasi-posterior converges to a normal distribution and exhibits strong connections with the loss-likelihood bootstrap posterior.
  • The method demonstrates well-calibrated frequentist coverage and provides a consolidated method-of-moments estimator through the loss-scale parameter.

Conclusions:

  • Quasi-posterior distributions provide a robust and theoretically sound Bayesian approach for generalized linear models.
  • The proposed method enhances reliability by relaxing strict distributional assumptions, making it suitable for real-world data.
  • The loss-scale parameter offers a meaningful interpretation of dispersion, integrating estimation and model assessment.