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Parametrization, Prior Independence, and the Semiparametric Bernstein-von Mises Theorem for the Partially Linear Model

Christopher D. Walker

TL;DR

This paper develops a Bernstein–von Mises theorem for the partially linear model by adopting a feasible adaptive parametrization based on the Robinson transformation, which separates the low-dimensional target $\beta$ from the infinite-dimensional nuisance $m=(m_1,m_2)$. By working in the $(\beta,m)$-parametrization, the author avoids the prior invariance condition that typically arises from information loss in semiparametric models and proves that the marginal posterior of $\beta$ is asymptotically normal with a semiparametric efficiency-bound variance. The theory is verified for two priors: uniform wavelet-series priors on $m$ and Matérn Gaussian process priors, with no cross-restrictions on the nuisance functions’ smoothness in the wavelet case and explicit regularity conditions in the Matérn case. The results imply that Bayesian credible sets for $\beta$ are asymptotically efficient frequentist confidence intervals and that parametrization choice materially affects feasibility and robustness of BVM results in semiparametric models.

Abstract

I prove a semiparametric Bernstein-von Mises theorem for a partially linear regression model with independent priors for the low-dimensional parameter of interest and the infinite-dimensional nuisance parameters. My result avoids a challenging prior invariance condition that arises from a loss of information associated with not knowing the nuisance parameter. The key idea is to employ a feasible reparametrization of the partially linear regression model that reflects the semiparametric structure of the model. This allows a researcher to assume independent priors for the model parameters while automatically accounting for the loss of information associated with not knowing the nuisance parameters. The theorem is verified for uniform wavelet series priors and Matérn Gaussian process priors.

Parametrization, Prior Independence, and the Semiparametric Bernstein-von Mises Theorem for the Partially Linear Model

TL;DR

This paper develops a Bernstein–von Mises theorem for the partially linear model by adopting a feasible adaptive parametrization based on the Robinson transformation, which separates the low-dimensional target from the infinite-dimensional nuisance . By working in the -parametrization, the author avoids the prior invariance condition that typically arises from information loss in semiparametric models and proves that the marginal posterior of is asymptotically normal with a semiparametric efficiency-bound variance. The theory is verified for two priors: uniform wavelet-series priors on and Matérn Gaussian process priors, with no cross-restrictions on the nuisance functions’ smoothness in the wavelet case and explicit regularity conditions in the Matérn case. The results imply that Bayesian credible sets for are asymptotically efficient frequentist confidence intervals and that parametrization choice materially affects feasibility and robustness of BVM results in semiparametric models.

Abstract

I prove a semiparametric Bernstein-von Mises theorem for a partially linear regression model with independent priors for the low-dimensional parameter of interest and the infinite-dimensional nuisance parameters. My result avoids a challenging prior invariance condition that arises from a loss of information associated with not knowing the nuisance parameter. The key idea is to employ a feasible reparametrization of the partially linear regression model that reflects the semiparametric structure of the model. This allows a researcher to assume independent priors for the model parameters while automatically accounting for the loss of information associated with not knowing the nuisance parameters. The theorem is verified for uniform wavelet series priors and Matérn Gaussian process priors.
Paper Structure (33 sections, 15 theorems, 303 equations)

This paper contains 33 sections, 15 theorems, 303 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Related Literature
  4. Outline of Paper
  5. Data Generating Process and Bayesian Inference
  6. Data Generating Process
  7. Bayesian Inference
  8. Bernstein-von Mises Theorem
  9. General Theorem
  10. Sufficient Conditions and Examples
  11. Useful Preliminary Results
  12. Example 1 -- Uniform Wavelet Series Priors
  13. Example 2 -- Matérn Gaussian Process Priors
  14. Discussion
  15. Why Theorem \ref{['thm:BVM']} Works
  16. ...and 18 more sections

Key Result

Theorem 1

Let $\tilde{I}_{n}(m_{0})$ be given by and let $\tilde{\Delta}_{n,0}$ be given by where If Assumptions as:dgp, as:prior, as:consistency, and as:emp_process hold and $\beta_{0} \in \text{int}(\mathcal{B})$, then for $P_{0,W}^{\infty}$-almost every fixed realization $\{w_{i}\}_{i \geq 1}$ of $\{W_{i}\}_{i \geq 1}$ as $n\rightarrow \infty$, where $||\cdot||_{TV}$ is total variation distance.The t

Theorems & Definitions (42)

  • Remark 1: Posterior Computation
  • Theorem 1
  • Remark 2: Unconditional Asymptotics
  • Corollary 1
  • Remark 3: Efficiency
  • Remark 4: Uncertainty Quantification
  • Corollary 2
  • Proposition 1
  • Remark 5: Relaxing Part 2 of Proposition \ref{['prop:suffgeneral']}
  • Proposition 2
  • ...and 32 more