I have a regression task, for which I'm training a model with MSE loss. So for label $y$ and estimation $\hat{y}$ the loss is $$\ell(y,\hat{y})=(y-\hat{y})^2$$ However, there is an uncertainty in the "true" labels, which varies across labels. So each true label is drawn from a distribution for which I can obtain a reasonable estimate for any statistic e.g. the standard deviation.
I'd like the loss to reflect the variation in the true label $y$. I thought about simply normalizing by the standard deviation of each label
$$\ell\left(y,\hat{y}\right)=\left(\frac{y-\hat{y}}{\sigma\left(y\right)}\right)^{2}$$
Or, since sometimes $\sigma(y)=0$, maybe
$$\ell\left(y,\hat{y}\right)=\left(\frac{y-\hat{y}}{1+\sigma\left(y\right)}\right)^{2}$$
But this seems too ad-hoc. Is there a standard theory or approach people use in this sort of situation?