I've got set of vectors, x_i, each of which is a fit to some data vectors. I know the covariance of those x_i, C_i, which I can infer from the noise on the original data. To be clear, these covariance matrices are not diagonal... they parameters inside each x_i do covary. However, I don't necessarily expect that the difference between those data vectors is entirely explicable by noise--there is likely also some unknown systematic that varies with i.

If I want to write down a sample mean xbar, I can do that as (∑ C_i^-1) (∑ C_i^-1 x_i). However, I'm confused about how to write down the unbiased sample covariance of x_i.

The closest thing I could find online was this post, but here the weights are scalars and not full inverse covariances.

Any help would be greatly appreciated. Thanks so much!