If you want to give the various datapoints weights

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of the individual datapoints beforehand. If you want to give the various datapoints weights based on something other than σ meas,m , then that is just like chi-square fitting except that you can adopt an arbitrary scale factor for the uncertainties (section 8.5). Chi-square fitting treats uncertainties of the derived parameters in a surprising way. Getting the coefficient uncertainties with chi-square fitting is a tricky business because
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– 24 – 1. With the standard treatments, the errors in the derived parameters don’t depend on the residuals of the datapoints from the fit (!). 2. The errors in the derived parameters can depend on their mutual covariances. This discussion requires a separate section, which we provide below in § 9. In this section we treat chi-square fitting ignoring covariance. We begin by illustrating the difference between least squares and chi-square fitting by discussing the simplest chi-square fitting case of a weighted mean; then we generalize to the multivariate chi-square fitting case. 8.1. The weighted mean: the simplest chi-square fit First, recall the formulas for an ordinary unweighted average in which the value of each point is y m and the residual of each point from the weighted mean is δy m : mean = y m M (8.2a) s 2 = δy 2 m M 1 (8.2b) s 2 mean = s 2 M = δy 2 m M ( M 1) , (8.2c) where s 2 mean is the variance of the mean and s 2 is the variance of the datapoints around the mean. Recall that in this case the mean is the least-squares fit to the data, so to use least squares jargon we can also describe s mean as the error in the derived coefficient for this single-parameter least-squares fit. Now for a weighted average in which the weight of each point is w meas,m = 1 σ 2 meas,m . Applying maximum likelihood, in an unweighted average the quantity that is minimized is δy 2 m ; in a weighted average the quantity minimized is χ 2 = δy 2 m σ 2 meas,m = w meas,m δy 2 m w meas,m δy 2 m , where to the right of the arrow we assume all w meas,m are identical. So your intuition says that the three equations corresponding to the above would become mean w,intuit = w meas,m y m w meas,m y m M (8.3a) Again, to the right of the arrow we assume all w meas,m are identical and the subscript intuit means “intuitive”. For the variances the intuitive expressions are s 2 w,intuit = M M 1 w meas,m δy 2 m w meas,m = hatwider χ 2 ( w meas,m /M ) δy 2 m M 1 = s 2 (8.3b)
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– 25 – s 2 w,mean,intuit = s 2 w,intuit M = w meas,m δy 2 m ( M 1) w meas,m = hatwider χ 2 w meas,m δy 2 m M ( M 1) = s 2 M . (8.3c) In fact, after a formal derivation, the first two equations (8.3a and 8.3b) are correct, so we will drop the additional subscipts intuit and formal on mean and s 2 w . However, after a formal derivation, the last of these equations becomes, and is always written (e.g. BR equation 4.19; Taylor equation 7.12) s 2 w,mean,formal = 1 w meas,m σ 2 meas M . (8.4) This is a problem , for the following reason.
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