Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Csiszár indices and interpolating copulas

Cristina Butucea, Jean-François Delmas, Anne Dutfoy, Antoine Schoonaert

Abstract

We study various properties of $f$-divergences and Csiszár indices between two probability distributions in very general setups for the convex function $f$ and for the probability distributions. We establish general structural properties of $f$-divergences and show how they are inherited by the associated Csiszár indices, including monotonicity and invariance under suitable transformations. We also study the relationship between Csiszár indices and copula representations of random vectors. When the marginal distributions have atoms, the copula representation is not unique and the Csiszár index of the transformed vectors may increase. We build a large family of interpolating copulas which minimize the Csiszár index and thus preserve the dependence structure of the initial vector.

Csiszár indices and interpolating copulas

Abstract

We study various properties of -divergences and Csiszár indices between two probability distributions in very general setups for the convex function and for the probability distributions. We establish general structural properties of -divergences and show how they are inherited by the associated Csiszár indices, including monotonicity and invariance under suitable transformations. We also study the relationship between Csiszár indices and copula representations of random vectors. When the marginal distributions have atoms, the copula representation is not unique and the Csiszár index of the transformed vectors may increase. We build a large family of interpolating copulas which minimize the Csiszár index and thus preserve the dependence structure of the initial vector.

Paper Structure

This paper contains 13 sections, 15 theorems, 93 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. On convexity
  4. On measures
  5. The $f$-divergence
  6. The Csiszár index
  7. Introduction
  8. Invariance of the Csiszár index
  9. Csiszár index and copulas
  10. Additional results and proofs
  11. On convex functions
  12. Proofs of results in Section \ref{['sec:csiszar_divergence']}
  13. Proofs of results in Section \ref{['sec:csiszar_index']}

Key Result

Lemma 1

Let $f\in {\mathbb F}$ (and thus $f(1)=0$) and $P,Q\in {\mathcal{P}}$. We have: $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Comparison of the copulas $C^\mathrm{cb}_Z$ and $C^\mathrm{ip}_Z$ in Example \ref{['exple:unique-cop-rep']}.
  • Figure 2: Two copulas for the discrete random variable $(X,Y)$ with Bernoulli marginals from Example \ref{['exple:different_copula_give_different_index']}.

Theorems & Definitions (47)

  • Remark 1
  • Definition 1
  • Remark 2: Domain of $f\in {\mathbb F}$
  • Remark 3: $f\in {\mathbb F}$ affine
  • Remark 4
  • Lemma 1: Properties of the $f$-divergence
  • Remark 5: Restriction to dominated probability measures
  • Remark 6: Examples of $f$-divergence
  • Remark 7: $f_\alpha$-divergence and Rényi divergence
  • Proposition 2: Invariance of the $f$-divergence
  • ...and 37 more