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 $x_{0}=1$, y values are independent of each other, 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.

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

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.

Assumptions

MLE has the following assumptions

• $x_0 = 1$
• $X$ is independent variable, and $y$ is dependent variable
• Residual error $\epsilon ~ N(0,\sigma^2)$ for a constant $\sigma^2$
• $y$ is independent across observations

Optimizations with log

Multiplication of many large numbers is computationally expensive. To optimize, the log of the likelihood function is computed. Since log of multiplied values is the sum of log of each value, we simplify the above down to

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

Loss function

Instead of maximizing the likelihood function, we can minimize the negative of the likelihood function. This way, we can just use OLS.