Update (4/3/2014): I believe I have solved the conjecture, and proven it to be correct. I will make a preprint available shortly(see link below). The original blog post remains available below. — Tom

Preprint available here: “An extremal inequality for long Markov chains“.

I have an extremal conjecture that I have been working on intermittently with some colleagues (including Jiantao Jiao, Tsachy Weissman, Chandra Nair, and Kartik Venkat). Despite our efforts, we have not been able to prove it. Hence, I thought I would experiment with online collaboration by offering it to the broader IT community.

In order to make things interesting, we are offering a $1000 prize for the first correct proof or counterexample! Feel free to post your thoughts in the public comments. You can also email me if you have questions or want to bounce some ideas off me.

Although I have no way of enforcing them, please abide by the following ground rules:

If you decide to work on this conjecture, please send me an email to let me know that you are doing so. As part of this experiment with online collaboration, I want to gauge how many people become involved at various degrees.
If you solve the conjecture or make significant progress, please keep me informed.
If you repost this conjecture, or publish any results, please cite this blog post appropriately.

One final disclaimer: this post is meant to be a brief introduction to the conjecture, with a few partial results to get the conversation started; it is not an exhaustive account of the approaches we have tried.

1. The Conjecture

Conjecture 1. Suppose ${X,Y}$ are jointly Gaussian, each with unit variance and correlation ${\rho}$ . Then, for any ${U,V}$ satisfying ${U-X-Y-V}$ , the following inequality holds:

$\displaystyle 2^{-2I(Y;U)} 2^{-2I(X;V|U)} \geq (1-\rho^2)+ \rho^2 2^{-2I(X;U)} 2^{-2I(Y;V|U)} . \ \ \ \ \ (1)$

2. Partial Results

There are several partial results which suggest the validity of Conjecture 1. Moreover, numerical experiments have not produced a counterexample.

Conjecture 1 extends the following well-known consequence of the conditional entropy power inequality to include long Markov chains.

Lemma 1 (Oohama, 1997). Suppose ${X,Y}$ are jointly Gaussian, each with unit variance and correlation ${\rho}$ . Then, for any ${U}$ satisfying ${U-X-Y}$ , the following inequality holds:

$\displaystyle 2^{-2 I(Y;U)} \geq 1-\rho^2+\rho^2 2^{-2I(X;U)}. \ \ \ \ \ (2)$

Proof: Consider any ${U}$ satisfying ${U-X-Y}$ . Let ${Y_u, X_u}$ denote the random variables ${X,Y}$ conditioned on ${U=u}$ . By Markovity and definition of ${X,Y}$ , we have that ${Y_u = \rho X_u + Z}$ , where ${Z\sim N(0,1-\rho^2)}$ is independent of ${X_u}$ . Hence, the conditional entropy power inequality implies that

$\displaystyle 2^{2h(Y|U)} \geq \rho^2 2^{2h(X|U)} + 2 \pi e(1-\rho^2) = 2 \pi e \rho^2 2^{-2I(X;U)} + 2 \pi e(1-\rho^2). \ \ \ \ \ (3)$

From here, the lemma easily follows. $\Box$

Lemma 1 can be applied to prove the following special case of Conjecture 1. This result subsumes most of the special cases I can think of analyzing analytically.

Proposition 1. Suppose ${X,Y}$ are jointly Gaussian, each with unit variance and correlation ${\rho}$ . If ${U-X-Y}$ are jointly Gaussian and ${U-X-Y-V}$ , then

$\displaystyle 2^{-2I(Y;U)} 2^{-2I(X;V|U)} \geq (1-\rho^2)+ \rho^2 2^{-2I(X;U)} 2^{-2I(Y;V|U)}. \ \ \ \ \ (4)$

Proof: Without loss of generality, we can assume that ${U}$ has zero mean and unit variance. Define ${\rho_u = E[XU]}$ . Since ${U-X-Y}$ are jointly Gaussian, we have

$\displaystyle I(X;U) =\frac{1}{2}\log\frac{1}{1-\rho_u^2} \ \ \ \ \ (5)$

$\displaystyle I(Y;U) =\frac{1}{2}\log\frac{1}{1-\rho^2\rho_u^2}. \ \ \ \ \ (6)$

Let ${X_u,Y_u,V_u}$ denote the random variables ${X,Y,V}$ conditioned on ${U=u}$ , respectively. Define ${\rho_{XY|u}}$ to be the correlation coefficient between ${X_u}$ and ${Y_u}$ . It is readily verified that

$\displaystyle \rho_{XY|u} = \frac{\rho\sqrt{1-\rho_u^2}}{\sqrt{1-\rho^2\rho_u^2}}, \ \ \ \ \ (7)$

which does not depend on the particular value of ${u}$ . By plugging (5)–(7) into (4), we see that (4) is equivalent to

$\displaystyle 2^{-2I(X;V|U)} \geq (1-\rho_{XY|u}^2)+ \rho_{XY|u}^2 2^{-2I(Y;V|U)}. \ \ \ \ \ (8)$

For every ${u}$ , the variables ${X_u,Y_u}$ are jointly Gaussian with correlation coefficient ${\rho_{XY|u}}$ and ${X_u-Y_u-V_u}$ form a Markov chain, hence Lemma 1 implies

$\displaystyle 2^{-2I(X_u;V_u)} \geq (1-\rho_{XY|u}^2)+ \rho_{XY|u}^2 2^{-2I(Y_u;V_u)}. \ \ \ \ \ (9)$

The desired inequality (8) follows by convexity of

$\displaystyle \log\left[(1-\rho_{XY|u}^2)+ \rho_{XY|u}^2 2^{-2z}\right] \ \ \ \ \ (10)$

as a function of ${z}$ . $\Box$

3. Equivalent Forms

There are many equivalent forms of Conjecture 1. For example, (1) can be replaced by the symmetric inequality

$\displaystyle 2^{-2(I(X;V)+I(Y;U))} \geq (1-\rho^2)2^{-2I(U;V)}+ \rho^2 2^{-2(I(X;U)+I(Y;V))}. \ \ \ \ \ (11)$

Alternatively, we can consider dual forms of Conjecture 1. For instance, one such form is stated as follows:

Conjecture 1′. Suppose ${X,Y}$ are jointly Gaussian, each with unit variance and correlation ${\rho}$ . For ${\lambda\in [{1}/({1+\rho^2}),1]}$ , the infimum of

$\displaystyle I(X,Y;U,V)-\lambda\Big(I(X;UV)+I(Y;UV)\Big), \ \ \ \ \ (12)$

taken over all ${U,V}$ satisfying ${U-X-Y-V}$ is attained when ${U,X,Y,V}$ are jointly Gaussian.

Here are some properties of the binary entropy function (bits) courtesy of Sergio Verdú. Please contribute any others you have found useful.

${0 \leq h(p) \leq 1\phantom{\frac{p}{p}}}$
${h(0.5) = 1 \mbox{~bit} \phantom{\frac{p}{p}}}$
${ h(p) = h(1-p) \phantom{\frac{p}{p}}}$
${h(0.11003) = 0.5 \mbox{~bits} \phantom{\frac{p}{p}}}$
${h \left( \frac{1}{k} \right) = \log_2 k - \left(1 - \frac{1}{k} \right) \log_2 (k-1), ~~k> 1}$
${\frac{d h(p)}{dp}|_{p=0} = \infty \phantom{\frac{p}{p}}}$
${\frac{d h(p)}{dp} = \log_2 \frac{1-p}{p}, ~~ 0 < p < 1}$
${\frac{d}{dp} \frac{h(p)}{1-p} = \frac{1}{(1-p)^2 }\log_2 \frac{1}{p} }$
${ \frac{h(p)}{1-p} ~\mbox{is monotonically increasing on} ~(0,1)}$
${p \log \alpha - h(p) ~\mbox{is monotonically decreasing/increasing on} }$ ${~\left(0, \frac{1}{1 + \alpha}\right) ~\mbox{and}~\left(\frac{1}{1 + \alpha}, 1 \right) ~ \mbox{respectively, for any}~\alpha > 0}$
${h(p) ~\mbox{is concave on} ~(0,1) \phantom{\frac{p}{p}}}$
${h(p) = p \log_2 \frac{e}{p} - \frac{\log_2 e}{2} p^2 + o(p^2) \phantom{\frac{p}{p}}}$
${\lim_{\alpha \downarrow 0} \frac{1}{\alpha} h \left( \frac{1}{1+\alpha} \right) - \log_2 \left( 1 + \frac{1}{\alpha} \right) = \log_2 e}$
${h \left( \frac{1+x}{2} \right) = 1 - \sum_{k=1}^\infty \frac{x^{2k}}{(2k-1) 2k} \log_2 e , ~~| x | \leq 1}$
${\lim_{n \rightarrow \infty} \frac{1}{n} \log_2 \binom{n}{k_n} = h (p ) ~\Longleftarrow ~\lim_{n \rightarrow \infty} \frac{k_n}{n} = p}$
${(1 - p q ) h \left( \frac{p - p q}{1 - p q } \right) = h(p) + p h(q) - h(pq) ~~0\leq p \leq 1, 0\leq q \leq 1, pq < 1 }$
${h(p) \leq 2 \sqrt{p (1-p)}\phantom{\frac{p}{p}}}$
${q h (p) \leq h ( q p ) , ~(p,q) \in [0,1]^2 \phantom{\frac{p}{p}}}$
${\int_0^{1} h ( x ) \, d x = \frac{1}{2} \log_2 {e}}$
${\int_0^1 \frac{h \left(x^\alpha\right)}{x} \,dx = \frac{\pi^2}{6 \alpha}\log_2 {e} }$
${\int_0^{\frac{\pi}{2}} h ( \sin^2 \alpha ) \, d\alpha = \frac{\pi}{2} \log_2 \frac{4}{e}}$
${\int_0^{1} u^{2k} h \left( \frac{1-u}{2}\right) \, du = \frac{\log_2 {e}}{(2k+1)(2k+2)} \sum_{i=0}^k \frac{1}{2i+1}~~~k=0,1,2, \ldots}$
${\max_{0 \leq \alpha \leq 1} h(\alpha ) + (1- \alpha) \beta_0 + \alpha \beta_1 = \log_2 ( 2^{\beta_0} + 2^{\beta_1} )}$
${h ( h^{-1} ( x ) * p ) ~\mbox{is convex on}~ 0<x<1 ~\mbox{for any}~ 0 \leq p \leq 1, \mbox{~where}}$ ${ ~ a * p = a (1-p) + p (1-a) \mbox{~and}~ h^{-1} ~\mbox{is the inverse of}~ h( x ) ~\mbox{on}~ \left[0, 1/2 \right] }$