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Bernstein von-Mises Theorem for g-prior and nonlocal prior

Xiao Fang, Malay Ghosh

Abstract

The paper develops Bernstein von Mises Theorem under hierarchical $g$ -priors for linear regression models. The results are obtained both when the error variance is known, and also when it is unknown. An inverse gamma prior is attached to the error variance in the later case. The paper also demonstrates some connection between the total variation and $α$-divergence measures.

Bernstein von-Mises Theorem for g-prior and nonlocal prior

Abstract

The paper develops Bernstein von Mises Theorem under hierarchical -priors for linear regression models. The results are obtained both when the error variance is known, and also when it is unknown. An inverse gamma prior is attached to the error variance in the later case. The paper also demonstrates some connection between the total variation and -divergence measures.
Paper Structure (11 sections, 8 theorems, 69 equations)

This paper contains 11 sections, 8 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Some Preliminary Result
  3. $\alpha$-divergence Criterion
  4. Tail bounds for the Chisquared distribution
  5. Notations
  6. Bernstein von-Mises theorem for g-prior
  7. Fixed $\sigma^2$ case
  8. Unknown $\sigma^2$ case
  9. Bernstein von-Mises theorem for nonlocal priors
  10. Discussion
  11. Proofs

Key Result

Lemma 1

Consider a sequence of densities $p_n(n\ge1))$ and a density $p$ all with respect to some $\sigma$-finite measure $\mu$. Then $TV(p_n,p)\to 0 \Rightarrow D_{\alpha}(p_n,p)\to 0$ for all $0<\alpha<1$.

Theorems & Definitions (9)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • Theorem 2
  • Theorem 3