If we find maximum likelihood of, $$ P(D_i\mid F_\text{true}) = \frac 1 {\sqrt{2\pi e_i^2}} \exp\left[ \frac{-(F_i - F_\text{true})^2}{2e_i^2} \right] $$ and calculate the $\log$ likelihood as: $$ \log\mathcal L = -\frac 1 2 \sum_{i=1}^n \left[ \log(2\pi e_i^2) + \frac{(F_i - F_\text{true})^2}{e_i^2} \right] $$ and calculate $f$ estimate as: $$ F_\text{est} = \frac{\sum w_i F_i}{\sum w_i}; \quad w_i = \frac 1 {e_i^2} $$ and then want to fitting a Gaussian approximation to the likelihood curve at maximum. then my question is how they calculate the standard deviation of this Gaussian approximation as: $$ \sigma_\text{est} = \left( \sum_{i=1}^N w_i \right)^{-1/2} $$
[References from this blog]