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On the large-sample limits of some Bayesian model evaluation statistics

Hien Duy Nguyen, Mayetri Gupta, Jacob Westerhout, TrungTin Nguyen

TL;DR

The paper develops a rigorous, general framework for the large-sample behavior of Bayesian information criteria—DIC, BPIC, and WBIC—under posterior consistency. It proves a consistency result for power posteriors $\Pi_n^{\beta_n}$ and a general a.s.-Lebesgue convergence theorem, enabling precise limit results for the criteria. Through three technical examples (geometric, normal, Laplace) and numerical experiments, it demonstrates when and how BPIC and WBIC (and DIC) converge to their asymptotic limits, clarifying the role of the tuning sequence $β_n$ (notably $nβ_n\to\infty$) and posterior concentration. The findings provide a principled, general perspective on the asymptotics of Bayesian model-evaluation criteria and offer a template for extending to broader Bayesian-consistency contexts, though they do not provide finite-sample guarantees or model-selection consistency in the classical sense.

Abstract

Model selection and order selection problems frequently arise in statistical practice. A popular approach to addressing these problems in the frequentist setting involves information criteria based on penalised maxima of log-likelihoods for competing models. In the Bayesian context, similar criteria are employed, replacing the maximised log-likelihoods with posterior expectations of the log-likelihood. Despite their popularity in applications, the large-sample behaviour of these criteria -- such as the deviance information criterion (DIC), Bayesian predictive information criterion (BPIC), and widely applicable Bayesian information criterion (WBIC) -- has received relatively little attention. In this work, we investigate the almost-sure limits of these criteria and establish novel results on posterior and generalised posterior consistency, which are of independent interest. The utility of our theoretical findings is demonstrated via illustrative technical and numerical examples.

On the large-sample limits of some Bayesian model evaluation statistics

TL;DR

The paper develops a rigorous, general framework for the large-sample behavior of Bayesian information criteria—DIC, BPIC, and WBIC—under posterior consistency. It proves a consistency result for power posteriors and a general a.s.-Lebesgue convergence theorem, enabling precise limit results for the criteria. Through three technical examples (geometric, normal, Laplace) and numerical experiments, it demonstrates when and how BPIC and WBIC (and DIC) converge to their asymptotic limits, clarifying the role of the tuning sequence (notably ) and posterior concentration. The findings provide a principled, general perspective on the asymptotics of Bayesian model-evaluation criteria and offer a template for extending to broader Bayesian-consistency contexts, though they do not provide finite-sample guarantees or model-selection consistency in the classical sense.

Abstract

Model selection and order selection problems frequently arise in statistical practice. A popular approach to addressing these problems in the frequentist setting involves information criteria based on penalised maxima of log-likelihoods for competing models. In the Bayesian context, similar criteria are employed, replacing the maximised log-likelihoods with posterior expectations of the log-likelihood. Despite their popularity in applications, the large-sample behaviour of these criteria -- such as the deviance information criterion (DIC), Bayesian predictive information criterion (BPIC), and widely applicable Bayesian information criterion (WBIC) -- has received relatively little attention. In this work, we investigate the almost-sure limits of these criteria and establish novel results on posterior and generalised posterior consistency, which are of independent interest. The utility of our theoretical findings is demonstrated via illustrative technical and numerical examples.

Paper Structure

This paper contains 37 sections, 11 theorems, 182 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General introduction
  3. Technical introduction
  4. Main results
  5. Consistency of Pinbetan
  6. A general convergence theorem
  7. Convergence of the BPIC and WBIC
  8. Convergence of the DIC
  9. Examples
  10. The geometric model
  11. The normal model
  12. The Laplace model
  13. Numerical examples
  14. DIC for the geometric model
  15. WBIC for the normal model
  16. ...and 22 more sections

Key Result

Proposition 1

Assume that $\left(X_{i}\right)_{i\in\mathbb{N}}$ are IID, $\Pi\left(\mathbb{B}_{\rho}\left(\theta_{0}\right)\right)>0$ for every $\rho>0$, and A1--A4 hold. If $n\beta_{n}\rightarrow\infty$, then the sequence of posterior measures $\left(\Pi_{n}^{\beta_{n}}\right)_{n\in\mathbb{N}}$ is consistent wit

Figures (2)

  • Figure 1: $\text{DIC}_{n}$ values for the geometric model. Points indicate the replicates of $\text{DIC}_{n}$ evaluations for a range of values of $\theta_{0}$ with different sample sizes $n$, while the lines correspond to the limiting value of $\text{DIC}_{n}$ as $n \to \infty$. Both points and lines are coloured according to the true value of $\theta_{0}$ in each case.
  • Figure 2: Simulated $\text{WBIC}_{n}$ values under various choices of $\beta_{n}$: (a) $1/\log n$, (b) $1/\log\log n$, (c) $1$, (d) $1/\sqrt{n}$, (e) $1/n$, and (f) $1/(n\log n)$. Points indicate the replicates of $\text{WBIC}_{n}$ evaluations, and the solid lines indicate the theoretical limit under Corollary \ref{['cor:-pbic-wbic-delta-convergence']}.

Theorems & Definitions (18)

  • Proposition 1
  • Theorem 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Theorem 2
  • proof
  • Proposition 4
  • proof
  • Remark 1
  • ...and 8 more