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The e-posterior.

Peter D Grünwald1,2

  • 1Machine Learning Group, CWI, Amsterdam, The Netherlands.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 27, 2023
PubMed
Summary
This summary is machine-generated.

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This study introduces e-posteriors for decision-making under uncertainty, offering frequentist valid risk bounds. E-posterior minimax rules are safer than Bayesian ones, especially when prior information is inadequate.

Area of Science:

  • Statistics
  • Decision Theory
  • Bayesian Inference

Background:

  • Traditional Bayesian inference relies on prior adequacy for valid results.
  • Frequentist methods offer validity but often lack predictive power for specific loss functions.
  • Unifying Bayesian and frequentist approaches remains a challenge.

Purpose of the Study:

  • To develop a novel representation of decision-maker uncertainty using e-variables.
  • To establish an 'e-posterior' that provides frequentist valid risk bounds.
  • To demonstrate the safety and applicability of e-posterior minimax decision rules.

Main Methods:

  • Development of e-variables to represent uncertainty.
  • Construction of the e-posterior, analogous to the Bayesian posterior.
Keywords:
Bayes-frequentist debateSavage–Dickey ratioconditional inferencedecision makinge-valuesuncertainty quantification

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  • Application of e-posteriors to re-interpret Kiefer-Berger-Brown-Wolpert conditional frequentist tests.

    • Main Results:

    • The e-posterior allows predictions for arbitrary loss functions.
    • Risk bounds derived from e-posteriors possess frequentist validity, independent of prior adequacy.
    • Inadequate e-collections lead to loose, not incorrect, risk bounds, enhancing safety.

    Conclusions:

    • E-posterior minimax decision rules offer a safer alternative to Bayesian rules.
    • The quasi-conditional paradigm unifies aspects of Bayesian and frequentist statistics.
    • This framework provides a robust approach to decision-making under uncertainty.