If we have N independent observations <a href="https://www.codecogs.com/eqnedit.php?latex=(\underline{x}_1,&space;y_1),&space;...,&space;(\underline{x}_N,&space;y_N)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?(\underline{x}_1,&space;y_1),&space;...,&space;(\underline{x}_N,&space;y_N)" title="(\underline{x}_1, y_1), ..., (\underline{x}_N, y_N)" /></a>, we can estimate the value of <ins>θ</ins> by maximizing the log-likelihood. We can optionally choose to weight some observations more or less that others by choosing weights <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{100}&space;w_i,&space;i&space;=&space;1,&space;...,&space;n" target="_blank"><img src="https://latex.codecogs.com/svg.latex?\dpi{100}&space;w_i,&space;i&space;=&space;1,&space;...,&space;n" title="w_i, i = 1, ..., n" /></a> and assuming that <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{100}&space;\small&space;y_i&space;\sim&space;N(f(\underline{x}_i,&space;\underline{\theta}),&space;\frac{\sigma^2}{w_i})" target="_blank"><img src="https://latex.codecogs.com/svg.latex?\dpi{100}&space;\small&space;y_i&space;\sim&space;N(f(\underline{x}_i,&space;\underline{\theta}),&space;\frac{\sigma^2}{w_i})" title="\small y_i \sim N(f(\underline{x}_i, \underline{\theta}), \frac{\sigma^2}{w_i})" /></a> for all i (where σ is unknown).
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