There is a vector $\vec{x}$ of length $N$, consisting of ones and zeroes. Initially $\sum_i x_i = N^x_I$. Now, each value of $\vec{x}$ that is equal to zero flips to $1$ with probability $p$. Let's call the resulting values $\vec{x}'$. The sum after this operation is $\sum_i x_i' = N^x_F$.
There is also a vector $\vec{y}$ of length $N$, which behaves analogously.
Also, define the dot products $D = \vec{x}\cdot\vec{y}$, $D' = \vec{x}'\cdot \vec{y}'$
Question: Estimate MLE and its variance of $D$, given $p, N, N^x_F, N^y_F, D'$. Note that nothing is known about initial values of $\vec{x}$ and $\vec{y}$, and all random processes are IID
Edit: As requested, I would like to clarify the origin of this problem. This is a minimal example of an actual problem in data estimation I am facing. I am trying to estimate the number optical channels that stay active before and after a certain operation has been performed. I know in advance that my measurement of the number of channels active at a given time is noisy, as I may falsely estimate some channels to be active when they are in fact not active. The opposite is not true - I will never underestimate the numeber of active channels.