But that form is not required. A quick counter example. I'm trying to estimate a value from N measurements. The measurements experience Gaussian noise with some

But that form is not required. A quick counter example. I'm trying to estimate a value from N measurements. The measurements experience Gaussian noise with some general covariance matrix K (i.e., they are not independent) Therefore, y is a sample from N([1, 1, ..., 1]^T u, K). The MLE is then ([1, 1, ..., 1]K^{-1}[1, 1, ..., 1]^T)^{-1} [1, 1, ..., 1] K^{-1} y. Or in words, multiply y by the inverse covariance matrix, sum the result, and divide by the sum of all the elements in the inverse covariance matrix. As a sanity check, when the measurements _are independent, this reduces to a weighted average, where the observations are weighted by their inverse variances.