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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon} is the residual error/noise

We assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0 = 1} , y values are independent of each other, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)}

The likelihood of observing the data is the product of observing each data point, given by the probability density function of standard distribution.

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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