How do you take the variance of the datapoints into account to compute variance estimates of the parameters of a non-linear least squares fit?

Suppose, I fit a non-linear model to a dataset $\mathbf{d} = [d_k]$, where each data point has a known variance $\mathrm{Var}\, d_k$. With the standard equation to compute the variance-covariance matrix, only the mean square error $\mathrm{MSE}$ and the Jacobian $\mathbf{J}$ are taken into account

$$\mathrm{Cov}\, p = \mathrm{MSE}\, \mathbf{J}^T \mathbf{J} .$$

The variance of the data points does not contribute. With weighted non-linear least squares, the relative weight of the datapoints is taken into account, but again, the variance of the parameters is independent of the "absolute" variance of the datapoints.

Am I missing something?


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