Talk:Inverse-variance weighting
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Confusing symbols in the Multivariate case section edit
The notation used in the section on the multivariate case is quite confusing, in the that the is used to indicate both a sum and a covariance matrix. Additionally, the symbol is used to denote a covariance matrix, whereas in the rest of the article is used to mean variance.
I have boldly edited the equations to use the more common symbol for covariance matrices. The older formulae are retained below.
of the individual estimates :
Derivation from maximum likelihood? edit
Let there be a set of measurements , each with uncertainty , of a variable . A "gaussian" probability distribution function of with respect to each measurment is:
The log-likelihood of given the measurements, ( could be multiplied with -1, doesn't matter):
Finding that maximizes likelihood should give the "best" estimator of the weighted-mean of the values, taking the uncertainties into account:
So then, from the above the "best" is:
Decomposing the variance of , we get: