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Cross posted to theory exchange - https://cstheory.stackexchange.com/questions/45610/bounding-information-of-expression

Suppose $u_1,\ldots,u_n$ are uniformly iid in $\{0,1\}$.

Let $x_1,\ldots,x_n$ be random variables taking values in $\{0,1\}$.

I'm trying to bound the following sum which expresses each $x_i$ has to decide how to distribute his information;

$\sum_i I(u_i ;x_i) + \sum_i I(u_i ; x_{<i})$ where $x_{<i}$ denotes $x_1,\ldots,x_{i-1}$. I want to prove this is at most $(2-c)n$ for some $c>0$.

I naively thought at first it's even possibly to bound the following-

Denote by $x^i$ the vector of all $x_j$ apart from $x_i$.

$$\sum_i I(u_i;x_i) + \sum_i I(u_i;x^i)$$

Here is why naive proofs don't work for the strengthed version-

If we try to do something similiar to $\sum_i I(u_i;x_j) \leq I(u^j;x_j) \leq H(x_j)$, a naive attempt is to try and split each expression $I(u_i; x^i) = \sum_{j\neq i} I(u_i;x_j \mid x_{<j,\neq i})$.

Then rearrange this sum by fixing the index of $x; j$ and running over $u_i$, and then hope that since the $u_i$ are independent, for any event $W$ we'd have $\sum_i I(u_i; x_j \mid W)$ is small. Sadly this seems to not be true, if you take $x_j$ to be random idd of the $u_i$, and $W$ the indexes where $x_j = u_i$, then this sum is $n$! Of course here we have powerful limitations on $W$, but I can't find a way to express them correctly.

Here is even a counter-example for the strong version. Consider for $i\neq 1$, $x_i=u_i$, and $x_1 = u_2 + u_3+ \cdots +u_n$ where the sum is $\bmod 2$.

Thus we somehow need to use the $x_{<i}$ vs $x^i$.

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  • $\begingroup$ Does $I$ denote conditional entropy? $\endgroup$ – LeechLattice Sep 28 at 4:44
  • $\begingroup$ @Bullet51 It denotes the common information. $I(x;y)=H(x)-H(x|y)$ $\endgroup$ – Andy Sep 28 at 8:30
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    $\begingroup$ Please do not post the same question on multiple sites. $\endgroup$ – D.W. Sep 28 at 16:26
  • $\begingroup$ @D.W. I cheked out that thread before I cross posted, and saw people going for each direction, so I felt this is okay. Moreover, I really believe my question is very relevant on both sites, and the fact it got no attention here (in upvotes\answers\comments) caused me to do it. $\endgroup$ – Andy Sep 28 at 16:39
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While no one apparently is interested, I finally got it and will sketch it here if in the future someone will be interested in this.

So we know

$\sum_1^n I(u_i;x_i) + \sum_2^n I(u_i; x_{<i})$ is large, say at least $(2-\epsilon)n$ I will allow myself during the proof to subtract consts from the left and still say it's at least $(2-\epsilon)n$, which is okay because we can slightly change $\epsilon$.

I will use $X$ for $(x_1 ,..,x_n)$, and $x_{<i}$ for $(x_1 ,..,x_{i-1})$

Lemma 1:

Let $A,B,C$ be random variables from a common distribution space, $I(A;C) \geq I(A;B)+I(B;C) - H(B)$

The proof of this lemma is omitted.

We can use it as follows, $\sum_2^n I(x_i ;x_{<i}) \geq \sum_2^n I(u_i;x_i) + \sum_2^n I(u_i; x_{<i}) - (n-2) \geq (1-\epsilon)n$

Lemma 2:

$\sum_2^n I(x_i ;x_{<i}) \geq (1-\epsilon)n$ implies

$H(X) \leq \epsilon n$ .

Proof of lemma 2 : Indeed $\sum_2^n I(x_i ;x_{<i}) = \sum_2^n H(x_i) - \sum_2^n H(x_i |x_{<i})$

Now add and subtract $H(X_1)$ and use the chain rule to get

$I(x_i ;x_{<i}) \leq \sum_1^n H(x_i) - H(X) \leq n - H(X)$

Concluding the proof: We find $H(X) \leq \epsilon$, but we have $\sum_1^n I(x_i;u_i) \geq (1-\epsilon)n$. This is a contradiction for $\epsilon < 1/2$ because $\sum_1^n I(x_i;u_i) \leq I(X;u_i) \leq I(X;U) \leq H(X)$

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