Show analytically that for scenario 1 (joint likelihood for the distances is Gaussian), the least-squares estimator of the position is equivalent to the maximum likelihood estimator.
Scenario 1: Measurements of all anchors follow the Gaussian model.
I have to prove that this equation: $ \widehat{\textbf{p}}ML = \underset{p}{\mathrm{argmax}} $p$ (\textbf{r}|\textbf{p})= \underset{p}{\mathrm{argmax}} \prod_{i=i}^{N_{a}} p(r_{i}|\textbf{p}) $
equals to this: $ \widehat{\textbf{p}}ML \approx \widehat{\textbf{p}}LS =\underset{p}{\mathrm{argmin}} \sum_{i=1}^{N_{a}} ((r_{i} - d_{i}(\textbf{p}))^2 = ||\textbf{r} - \textbf{d}(\textbf{p})||^2 $
The dependence on the parameter p is given as the Euclidean distance to the i-th anchor$$ d_{i}(\textbf{p}) = \sqrt((x_{i} - x)^2(y_{i} - y)^2)$$
Thanks for any help!