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Asymptotics for power posterior mean estimation

Ruchira Ray, Marco Avella Medina, Cynthia Rush

Abstract

Power posteriors "robustify" standard Bayesian inference by raising the likelihood to a constant fractional power, effectively downweighting its influence in the calculation of the posterior. Power posteriors have been shown to be more robust to model misspecification than standard posteriors in many settings. Previous work has shown that power posteriors derived from low-dimensional, parametric locally asymptotically normal models are asymptotically normal (Bernstein-von Mises) even under model misspecification. We extend these results to show that the power posterior moments converge to those of the limiting normal distribution suggested by the Bernstein-von Mises theorem. We then use this result to show that the mean of the power posterior, a point estimator, is asymptotically equivalent to the maximum likelihood estimator.

Asymptotics for power posterior mean estimation

Abstract

Power posteriors "robustify" standard Bayesian inference by raising the likelihood to a constant fractional power, effectively downweighting its influence in the calculation of the posterior. Power posteriors have been shown to be more robust to model misspecification than standard posteriors in many settings. Previous work has shown that power posteriors derived from low-dimensional, parametric locally asymptotically normal models are asymptotically normal (Bernstein-von Mises) even under model misspecification. We extend these results to show that the power posterior moments converge to those of the limiting normal distribution suggested by the Bernstein-von Mises theorem. We then use this result to show that the mean of the power posterior, a point estimator, is asymptotically equivalent to the maximum likelihood estimator.
Paper Structure (11 sections, 4 theorems, 65 equations)

This paper contains 11 sections, 4 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Notation and preliminaries
  4. Assumptions
  5. Convergence of power posterior moments
  6. Asymptotic normality of the Bayes estimator
  7. Discussion
  8. Proofs of Main Results
  9. Proof of Theorem \ref{['thm:moments_alt']}
  10. Proof of Theorem \ref{['thm:BayesEstimator']}
  11. Technical Lemmas

Key Result

Theorem 1

Assume (A0)--(A3) hold. Then, for any integer $k\in[1,k_{0}]$, in $f_{0,n}$-probability, where is the $1$-norm of a $k^{th}$ order tensor and are the centered and scaled versions of the $\alpha$-posterior and its limiting normal.

Theorems & Definitions (9)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Lemma 1
  • Proof
  • Lemma 2
  • Proof