http://www.itsoc.org/about/shannons-centenary-us-postal-stamp

Recently I have encountered an inequality involving a product of mutual informations, which I cannot find a good way to prove (or disprove, though some numerics and asymptotic analysis seem to suggest its validity). I would much appreciate it if someone could be smart and gracious enough to provide a proof, or counter example, or generalization.

The formulation is quite simple: suppose is a binary random variable with  . The random variable is the output of passing through a binary symmetric channel. Moreover is another random variable such that forms a Markov chain. The claim is that . (Note that the left side is also upper bounded by either one of the two factors on the right side, by the well-known data processing  inequality.

At first glance this inequality seems absurd because different units appear on the two sides; but this may be resolved by considering that has only one bit of information.

For a given joint distribution of and , the equality can be achieved when is an injective function of .

In order to make things interesting, we are offering a $1000 prize for the first correct proof of the conjecture, or a $250 award for the first 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:

1. 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.
2. If you solve the conjecture or make significant progress, please keep me informed.
3. 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 are jointly Gaussian, each with unit variance and correlation . Then, for any satisfying , the following inequality holds:

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 are jointly Gaussian, each with unit variance and correlation . Then, for any satisfying , the following inequality holds:

Proof: Consider any satisfying . Let denote the random variables conditioned on . By Markovity and definition of , we have that , where is independent of . Hence, the conditional entropy power inequality implies that

From here, the lemma easily follows.

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 are jointly Gaussian, each with unit variance and correlation . If are jointly Gaussian and , then

Proof: Without loss of generality, we can assume that has zero mean and unit variance. Define . Since are jointly Gaussian, we have

Let denote the random variables conditioned on , respectively. Define to be the correlation coefficient between and . It is readily verified that

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

For every , the variables are jointly Gaussian with correlation coefficient and form a Markov chain, hence Lemma 1 implies

The desired inequality (8) follows by convexity of

as a function of .

3. Equivalent Forms

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

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

Conjecture 1′. Suppose are jointly Gaussian, each with unit variance and correlation . For , the infimum of

taken over all satisfying is attained when are jointly Gaussian.

http://www.whitehouse.gov/photos-and-video/video/2013/02/01/president-obama-honors-countrys-top-innovators-and-scientists-2011

for outstanding contributions in Mathematical Aspects of Information Sciences including:

1. All mathematical aspects of computer science, including complexity theory, logic of programming languages, analysis of algorithms, cryptography, computer vision, pattern recognition, information processing and modelling of intelligence.
2. Scientific computing and numerical analysis. Computational aspects of optimization and control theory. Computer algebra.

Nomination Procedure: http://www.mathunion.org/general/prizes/nevanlinna/details/

http://imaiai.oxfordjournals.org/content/current

It includes the following editorial:

Statement 1. The “variability” of sample mean decreases with .

Statement 2. Let the average of functions be defined as . Then is less “variable” than .

To make these statements precise, one faces the fundamental question of comparing two random variables and (or more precisely comparing two distributions). One common way we think of ordering random variables is the notion of stochastic dominance:

However, this notion really is only a suitable notion when one is concerned with the actual size of the random quantities of interest, while, in our scenario of interest, a more natural order would be that which compares the variability between two random variables (or more precisely, again, the two distributions). It turns out that a very useful notion, used in a variety of fields, is due to Ross (1983): Random variable is said to be stochastically less variable than random variable (denoted by ) when every risk-averse decision maker will choose over (given they have similar means). More precisely, for random variables and with finite means

where is the set of functions for which the above expectations exist.

One interesting, but perhaps not entirely obvious, fact is that this notion of ordering is equivalent to saying that there is a sequence of mean preserving spreads that in the limit transforms the distribution of into the distribution of another random variable with finite mean such that ! Also, using results by Hardy, Littlewood and Polya (1929), the stochastic variability order introduced above can be shown to be equivalent to Lorenz (1905) ordering used in economics to measure income equality.

Now with this, we are ready to formalize our previous statements. The first statement is actually due to Arnold and Villasenor (1986):

Note that when you apply this fact to a sequence of iid random variables with finite mean , it strengthens the strong law of large number in that it ensures that the almost sure convergence of the sample mean to the mean value occurs with monotonically decreasing variability (as the sample size grows).

The second statement comes up in proving certain optimality result in sharing parallel servers in fork-join queueing systems (J. 2008) and has a similar flavor:

The cleanest way to prove both statements, to the best of my knowledge, is based on the following theorem first proved by Blackwell in 1953 (later strengthened to random elements in separable Banach spaces by Strassen in 1965, hence referred to by some as Strassen’s theorem):

Theorem 1 Let and be two random variables with finite means. A necessary and sufficient condition for is that there are two random variables and with the same marginals as and , respectively, such that almost surely.

For instance, to prove the first statement we consider and . All that is necessary now is to note that , is an independent uniform rv on the set , has the same distribution as random variable . Furthermore,

Similarly to prove the second statement, one can construct by selecting a random permutation of functions .