Talk:Inverse-variance weighting

Confusing symbols in the Multivariate case section

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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  :

 
  — Preceding unsigned comment added by Glopk (talkcontribs) 16:11, 8 September 2023 (UTC)Reply

Derivation from maximum likelihood?

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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:

    Blakut (talk) 08:52, 14 June 2023 (UTC)Reply