Maximum likelihood estimation

Maximum likelihood estimation (MLE) is one of the methods to find the coefficients of a model that minimizes the RSS in linear regression. MLE does this by maximizing the likelihood of observing the training data given a model.

Background

Consider objective function

$y=w_{0}x_{0}+w_{1}x_{1}+\ldots +w_{m}x_{m}+\epsilon =g(x)+\epsilon$

where $y=g(x)$ is the true relationship and $\epsilon$ is the residual error/noise

We assume that $x_{0}=1$, and $\epsilon \sim N(0,\sigma ^{2})$

Likelihood function

The likelihood function determines the likelihood of observing the data given the parameters of the model. A high likelihood indicates a good model.

$L(w_{i},\sigma ^{2}|x,y)=\prod {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}exp\left(-{\frac {(y_{i}-g(x_{i}))^{2}}{2\sigma ^{2}}}\right)$

For every data point, the likelihood is computed. The product of all likelihoods are taken.

The weights are then changed to fit it better, and the process repeats.

The computation can be simplified to the following

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