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The van Trees inequality in the spirit of Hajek and Le Cam

Gassiat Elisabeth, Stoltz Gilles

Abstract

We work out a version of the van Trees inequality in a Hajek--Le Cam spirit, i.e., under minimal assumptions that, in particular, involve no direct pointwise regularity assumptions on densities but rather almost-everywhere differentiability in quadratic mean of the model. Surprisingly, it suffices that the latter differentiability holds along canonical directions -- not along all directions. Also, we identify a (slightly stronger) version of the van Trees inequality as a very instance of a Cramer--Rao bound, i.e., the van Trees inequality is not just a Bayesian analog of the Cramer--Rao bound. We provide, as an illustration, an elementary proof of the local asymptotic minimax theorem for quadratic loss functions, again assuming differentiability in quadratic mean only along canonical directions.

The van Trees inequality in the spirit of Hajek and Le Cam

Abstract

We work out a version of the van Trees inequality in a Hajek--Le Cam spirit, i.e., under minimal assumptions that, in particular, involve no direct pointwise regularity assumptions on densities but rather almost-everywhere differentiability in quadratic mean of the model. Surprisingly, it suffices that the latter differentiability holds along canonical directions -- not along all directions. Also, we identify a (slightly stronger) version of the van Trees inequality as a very instance of a Cramer--Rao bound, i.e., the van Trees inequality is not just a Bayesian analog of the Cramer--Rao bound. We provide, as an illustration, an elementary proof of the local asymptotic minimax theorem for quadratic loss functions, again assuming differentiability in quadratic mean only along canonical directions.
Paper Structure (18 sections, 7 theorems, 58 equations, 1 figure)

This paper contains 18 sections, 7 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. One-dimensional version
  3. Statement
  4. Comparison to classic regularity assumptions
  5. Proof of Theorem \ref{['th:vT1']}
  6. Overview of the proof.
  7. Preparations.
  8. Proof of \ref{['eq:vT-infoeq']}.
  9. Conclusion by a Cauchy--Schwarz inequality.
  10. Special case.
  11. The van Trees inequality as a Cramér--Rao bound
  12. Multivariate version
  13. Comparison to classic regularity assumptions
  14. Statement
  15. Proof of Theorem \ref{['th:vT-multi']}
  16. ...and 3 more sections

Key Result

Theorem 4

The one-dimensional van Trees inequality eq:vT1 holds with $\mathcal{I}_{\mathbb{Q}} > 0$ under Assumption ass:def and the following additional assumptions: The first assumption holds in particular if the model $\mathcal{P}$ is differentiable in $\mathbb{L}^2(\mu)$ at all points of $\Theta \cap \mathrm{Supp}(q)$, not just almost everywhere.

Theorems & Definitions (15)

  • Definition 1: Differentiability in $\mathbb{L}_2$
  • Definition 2: Well-behaved prior
  • Theorem 4
  • Lemma 6: Pol01
  • proof
  • Corollary 7
  • Definition 8: nice functions
  • Definition 9: Differentiability in $\mathbb{L}_2$ along canonical directions
  • Definition 10: Well-behaved prior, multivariate version
  • Theorem 12
  • ...and 5 more