Let's assume $a, b \in \mathbb{R}^n$ are known, and $x \in [0, 1]^n$ is a random vector drawn uniformly from the unit cube. Given $y = a \cdot x$, the value of the dot product of $a$ with $x$, how can I estimate $b \cdot x$ in an unbiased way?
It seems that such an unbiased estimator should exist since we could just average the values of $b \cdot \hat x$ over all of the $\hat x \in [0, 1]^n$ with $a \cdot \hat x = y$, but I'm having trouble coming up with the formula / algorithm to calculate this estimate.