on Gaussian Correlation Inequality

Overview

The Gaussian Correlation Inequality was proved in 2014 (arXiv: 1408.1028), see (Royen, 2014). The interesting story can be found in Quanta Magazine.

🏷️ Introduction

The inequality is to show the Gaussian measure $\mu$ on centrally symmetric convex sets $A$ and $B$ satisfies

$$\mu (A\cap B)\ge \mu (A)\mu (B).$$

That is to say, if a dart hits the wall with standard Gaussian distribution, suppose two targets are centrally symmetric convex sets $A$, $B$, then hitting both targets with one dart is easier than hitting $A$ with the first dart and hitting $B$ with the second, vice versa.

The proof of inequality is simple and elegant. I think there are a few keys in the proof which are insightful (that is why this note exists 😁). The first observation is the following.

Observation

A centrally symmetric convex closed set can be formed by the intersection of countable symmetric strips (Schechtman, Schlumprecht & Zinn, 1998).

This observation of “strips” is natural, since a convex symmetric body can be approximated by a sequence of convex, symmetric polytopes. Moreover, convex, symmetric polytopes are just slices of the unit cube in a higher dimension satisfying the constraints $\{|⟨x,v_i⟩|\le 1\}$ for $i=1,2,\cdots ,k$. Therefore, the problem can be reduced to proving

Gaussian Correlation Theorem

$$\mathbb{P}\left(\bigcap _{i=1}^n{A}_i\right)\ge \mathbb{P}\left(\bigcap _{i=1}^k{A}_i\right)\mathbb{P}\left(\bigcap _{i=k+1}^n{A}_i\right),$$

where $A_i=\{|X_i|\le x_i\}$, $i=1,2,\cdots ,n$ and $(X_1,\cdots ,X_n)\sim \mathcal{N}(0,{\Sigma }_n)$.

The special case $k=1$ was proved by the following theorem (Khatri, 1967).

Theorem (Khatri)

Let $\{X_i{\}}_{1\le i\le n}$ be a jointly Gaussian random variables with mean zero. Then

$$\mathbb{P}(\underset{1\le i\le n}{\max }|X_i|\le 1)\ge \mathbb{P}(|X_1|\le 1)\mathbb{P}(\underset{2\le i\le n}{\max }|X_i|\le 1).$$

🏷️ Multivariate Gamma-type distribution

The set $\{|X_i|\le 1\}$ is better described by chi-squared distribution or Gamma distribution. Surprisingly, the multivariate Gamma distributions on ${\mathbb{R}}^n$ have several (non-equivalent) definitions🤣.

The $\Gamma (\alpha ,R)$ distribution is defined as follows.

Definition

If the random vector $X=(X_1,\cdots ,X_n)$ satisfies the Laplace transform

$$\mathbb{E}[\exp \left(-⟨s,X⟩\right)]=\frac{1}{|I+R\mathrm{diag}(s){|}^{\alpha }},$$

then it obeys the $\Gamma (\alpha ,R)$ distribution.

Example

If $\alpha =\frac{1}{2}$ and $X\sim {\chi }_n^2$ or $X\sim \Gamma (\alpha =\frac{n}{2},\theta =2)$, then the covariance matrix $R=2I_n$ and

$$\mathbb{E}[\exp (-⟨s,X⟩)]=\prod _{i=1}^n\mathbb{E}[\exp (-s_i{X}_i)]=\prod _{i=1}^n{\int }_0^{\infty }\frac{x^{-1/2}{e}^{-x/2}}{2^{1/2}\Gamma (\frac{1}{2})}{e}^{-s_{i}x}dx=\prod _{i=1}^n\frac{1}{\sqrt{2s_i+1}}.$$

Note, not all values of $\alpha$ suffice to produce an admissible distribution. Some possible values of $\alpha$ are given in (Krishnamoorthy & Parthasarathy, 1951).

Example

Suppose $X_i\sim \mathcal{N}(0_p,R)$, then the Wishart matrix $S={\sum }_{i=1}^n{X}_i{X}_i^T\sim {\mathcal{W}}_p(n,R)$. The diagonal part of $S$ is ${\sum }_{i=1}^n{X}_i\odot X_i$ by Hadamard product. Then, the Laplace transform (or equivalently moment generating function) is

$$\begin{array}{rl}\mathcal{L}(\mathrm{diag}(S))& =\int d{X}_1\cdots d{X}_{n}p(X_1,X_2,\cdots ,X_n)e^{-s^T({\sum }_{i=1}^n{X}_i\odot X_i)}\\ & ={\left[\int dX\frac{1}{\sqrt{2\pi }}|R{|}^{-1/2}\exp \left(-\frac{1}{2}{X}^T{R}^{-1}X\right)\exp (-s^T(X\odot X))\right]}^n\\ & ={\left[\int dX\frac{1}{\sqrt{2\pi }}|R{|}^{-1/2}\exp \left(-\frac{1}{2}{X}^T(R^{-1}+2\mathrm{diag}(s))X\right)\right]}^n\\ & =|R{|}^{-n/2}|(R^{-1}+2\mathrm{diag}(s)){|}^{-n/2}\\ & =|I+2R\mathrm{diag}(s){|}^{-n/2}.\end{array}$$

Therefore, all $2\alpha \in \mathbb{N}$ are admissible values.

🏷️ Variational Technique

• In order to distinguish the dependence and independence, it is very common to introduce the correlation matrix for the $n$ dimensional vector $X$, that is, $C(\tau )=[C_{11},\tau C_{12};\tau C_{21},C_{22}]$ in the spirit of variational method, then the left-hand and right-hand sides of the desired inequality are referring the case $\tau =1$ and $\tau =0$. It equivalently means the function

$$\tau \mapsto \mu (Z_i(\tau )\le s_i,i=1,2\dots ,n)$$

is non-decreasing in $\tau$, where $Z_i=\frac{X_i^2}{2}$ and $s_i=\frac{t_i^2}{2}$. Let $f(z,\tau )$ be the joint distribution’s density function of $Z(\tau )$, then that is to show the derivative ${\partial }_{\tau }f(z,\tau )$ is non-negative.

• The following claim is from Lebesgue’s dominated convergence theorem. The differentiation can be swapped with the Laplace transform.

$${\int }_{[0,\infty {)}^n}{e}^{-\lambda \cdot \mathbf{z}}{\partial }_{\tau }f(z,\tau )dz={\partial }_{\tau }{\int }_{[0,\infty {)}^n}{e}^{-\lambda \cdot \mathbf{z}}f(z,\tau )dz$$

which equals to

$${\int }_{[0,\infty {)}^n}{e}^{-\lambda \cdot \mathbf{z}}f(z,\tau )dz=\mathbb{E}\exp (-\frac{1}{2}\sum _{i=1}^n{\lambda }_i{X}_i^2(\tau ))=|I+\Lambda C(\tau ){|}^{-1/2}.$$

• The rest is a linear algebra problem only. Here $\Lambda$ is not important anymore, we drop it as identity.

$$\det (I+C(\tau ))=\det (C_{11}+I)\det (C_{22}+I-{\tau }^2(C_{21})(C_{11}+I{)}^{-1}(C_{21}{)}^T)$$

which should be decreasing in $\tau$.

In the original proof by Thomas Royen, the inequality is extended to the distributions such that the Laplace transform is infinitely divisible.

🏷️ Further discussions

• If the convex sets are not quite symmetric (say up to some local perturbations), does the inequality still hold for Gaussian measure? Such question is raised naturally, some existing works on GCI for asymmetric sets (Cordero-Erausquin, 2002) (Lim & Luo, 2012) might be a starting point.

📚 References

🐻  Cordero-Erausquin, D. 2002. Some applications of mass transport to Gaussian-type inequalities. Archive for rational mechanics and analysis 161, 257–269.

🐻  Khatri, C.G. 1967. On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. The Annals of Mathematical Statistics, 1853–1867.

🐻  Krishnamoorthy, A.S. & Parthasarathy, M. 1951. A multivariate gamma-type distribution. The Annals of Mathematical Statistics 22(4), 549–557.

🐻  Lim, A.P.C. & Luo, D. 2012. A note on Gaussian correlation inequalities for nonsymmetric sets. Statistics & Probability Letters 82(1), 196–202.

🐻  Royen, T. 2014. A simple proof of the Gaussian correlation conjecture extended to multivariate gamma distributions.

🐻  Schechtman, G., Schlumprecht, T. & Zinn, J. 1998. On the Gaussian measure of the intersection. Annals of probability 26(1), 346–357.