I know that one way to estimate the weighted sum method is to use the inverse of each attribute's variance: $1/\sigma^2$. In wiki, it is saying: The significance of this choice is that this weighted mean is the maximum likelihood estimator of the mean of the probability distributions under the assumption that they are independent and normally distributed with the same mean.
How about weighted product? How do I estimate the power? suppose I have 2 measurement vector X and Y both range from 0 to 1 that I want to combine them together as: $z_i=x_i^{w_x}*y_i^{w_y}$. X and Y all follow normal distribution. I tested my data that $\sigma^2(X)=20$, $\sigma^2(Y)=53$. Can I use their ratio $\frac{20}{53}=\frac{1}{2.65}$ as their weight which is $z_i=x_i*y_i^{2.65}$? Or it is different for weighted product than weighted sum?