Weighted fitting and correlated errors

For independent measurements of one common mean mu with known variances, minimizing the sum of squared residuals divided by variance gives inverse-variance weights. Normalize the weights so they sum to one. This ensures the linear combination has expectation mu when the model is correct. The variance is the reciprocal of the total inverse variance. These conclusions require the quoted uncertainties to represent the sampling variances, rather than arbitrary plotting errors or weights chosen because a result looks plausible.

The equivalence to maximum likelihood uses Gaussian observations and fixed variances. Linear least squares can still define useful estimators without Gaussianity, but its exact normal-distribution interval and chi-square reference distribution no longer follow automatically. If the variance depends on the parameter, the Gaussian log determinant also depends on that parameter and must be included in likelihood fitting. Replacing every Poisson variance by its noisy observed count can similarly bias a naive weighted fit, especially when some bins are empty.

For a measurement vector y, let C be its known positive-definite covariance matrix and H the design matrix of a linear mean model. The generalized least-squares estimator follows by differentiating the covariance-weighted quadratic residual. Its covariance is obtained by propagating C through the resulting linear estimator. In practice solve linear systems or factor C; do not infer numerical stability merely from writing an inverse. Identifiability requires the relevant weighted design matrix to have full column rank.

A shared additive calibration uncertainty of variance tau squared contributes tau squared to every element of a covariance block when its sensitivity is one for each measurement. For repeated measurements of the same mean, this component does not average down. If the independent-error covariance is diagonal, adding this common component leaves the optimal normalized weights unchanged, but increases the variance of their average by tau squared. This special result should not be generalized to arbitrary correlation patterns or multiplicative uncertainties.

For two equal-variance readings, a positive covariance enlarges the variance of their mean but reduces that of their difference. A calibration offset can therefore cancel in a contrast even when it remains the dominant uncertainty of an absolute measurement. Negative correlation has the opposite effect. Reporting only individual standard deviations discards this distinction and can prevent a later user from reconstructing either quantity correctly. Correlation coefficients alone are also insufficient unless their associated standard deviations and ordering are supplied.

A weighted average assumes all inputs measure the same defined quantity. Measurements at different temperatures, energies or selection thresholds may require a response model before combination. An unusually large residual can indicate a statistical fluctuation, an underestimated covariance, or model mismatch; deleting it because it is inconvenient changes the procedure. Examine residual directions, calibration evidence and sensitivity to defensible covariance alternatives. A small quoted combination error is useful only when the common-quantity and uncertainty models remain credible.