on Gaussian Correlation Inequality

Info

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 (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.