I have $n$ **IID** Bernoulli random variables denoted by $X_1,X_2,\ldots X_n$ with parameter $p$.

I am interested in knowing if the following inequality involving mutual information holds :

$\boxed{\max_{p} I(X_1+X_2+...+X_n;Y)\leq \max_{p} I(2X_1+X_3+...+X_n;Y) \leq ...\leq \max_{p} I(nX_1;Y)}$

Note that $p$ is variable and can be different across different mutual information terms in the inequality.

Here, $Y$ is a binary random variable taking values in $\{0,1\}$. Also, $Y=0$ with probability $\frac{1}{x+5}$ when the input, ie, the first argument in the mutual information $I(X;Y)$ takes a value $x$.

**Note:** In fact, I conjecture that the inequality even holds for any decreasing transition probability in the place of $\frac{1}{x+5}$.

Can someone help me provide some insights or ways of proving these? In fact, my numerical simulations tend to agree with the inequality. Any help is appreciated.

**Numerical evidence:**

I considered $n=3$ as example. I found the following through simulation. $\max_{p} I(X_1+X_2+X_3;Y)=0.0027$ and is at $p=0.38$.

$\max_{p} I(2X_1+X_3;Y)=0.0043$ and is at $p=0.44$.

$\max_{p} I(3X_1;Y)=0.0075$ and is at $p=0.52$.

Also, when I considered a constant value of $p$ across all terms in the inequality, the inequality still seems to hold for the transition probability function $\frac{1}{x+5}$. But, it did not hold for other decreasing transition probability functions that I tried.

The original inequality with $\max_p$ still holds even for this different transition probability. So, I think the $\max_p$ is needed for the inequality to hold in the general sense.

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