Suppose I have random variables $A,B,C$ where $B$ is statistically-independent of $A$ and $C$ (e.g. it is Gaussian noise generated without regard for the other variables). Consider scalars $0<α<β<1$ and random variables $X=(1−α)A+αB$ and $Y=(1−β)A+βB$. Is it true that $I(C;X)≥I(C;Y)$ where $I$ denotes mutual information?
Intuitively I feel that this should be true because $X$ and Y are weighted sums of $A$ which may 'contain' information about $C$, and $B$ which does not. Since $X$ weights $A$ more-heavily and $B$ less-heavily relative to $Y$, it should 'contain' more information about $C$. However, I am not sure how to rigorously show this. Thanks in advance for any help.
Although very intuitive, there are some edge cases which ruin this in general. As a counterexample, consider $C = A \sim \mathrm{Unif}\{2,4\}$, $B \sim \mathrm{Unif}\{0,2\}$ and $\alpha = 0.5, \beta = 0.99$.
Then notice that $Y = \beta A + (1-\beta) B$ is "invertible": $Y \in \{ 0.02, 0.04, 2, 2.02\},$ and only one pair of values of $(A,B)$ realise each value of $Y$, and so upon seeing $Y$, you can figure out the realisations of $(A,B)$. This means that $I(A;Y) = H(A) = 1$.
However, $X$ is not "invertible" in this sense: $X \in \{ 1,2, 3\},$ and while $X = 1$ or $X = 3$ lets you figure out $A$, the case $X = 2$ is ambiguous, because you don't know if this arises because $(A,B) = (2,2)$ or $(4,0)$. So, $I(A;X) < H(A)$, yielding a counterexample.
A relation of this type that does hold is the following: let $U_1 \sim \mathrm{Bern}(\alpha)$ and $U_2 \sim \mathrm{Bern}(\beta)$ be independent of $(A,B,C)$. Then suppose that $X = (1-U_1)A + U_1 B,$ and $Y = (1-U_2)A + U_2 B$. In other words, rather than mixing the values of $A$ and $B$, we are mixing the distributions. In this case it does hold that $I(C;X) \ge I(C;Y),$ essentially due to the intuition you have (try to show this).
The original statement should also hold if we specialise to restricted classes of distributions (e.g., if $(A,B,C)$ were jointly Gaussian; show this too). If you need this for some application, then you should try to specify which class of laws you care about, and try to analyse the same.