Main Idea:

  1. Make an explicit assumption about what distribution the data was modeled from
  2. Set the parameters of this distribution so that the data we observe is most likely i.e maximize the likelihood of our data

For a simple example of a coin toss, we can see this as maximizing the probability of observing heads from a binomial distribution:

\[p(z_1, ..., z_n) = p(z_1 ...., z_n|\theta)\]

we assume I.I.D condition and so we should be able to break this down into :


Formally, let us deinfe a likelihood function as:

\[L(\theta) = \prod_{i=1}^N p(z_i|\theta)\]

Now, our task it to find the \(\hat{\theta}\) that maximizes this likelihood:

\[\hat{\theta}_{MLE} = \underset{\theta}{\text{argmax}} \prod_{i=1}^N p(z_i|\theta)\]

instead of maximizing a product, we can also view this problem as minimizing a sum if we take the log of all values:

\[\hat{\theta}_{MLE} = \underset{\theta}{\text{argmax}} \sum_{i=1}^N Log(p(z_i|\theta))\]

Let us use this idea for the regression problem. We assume that our outputs are distributed in a Gaussian manner around the line w have to find out. This basically means that our \(\epsilon\) is a Gaussian Noise that is messing up our outputs from the fundamental distribution

Thus, our equation for getting this probability of our output would be :

\[p(y_i|x_i; w_i, \sigma^2) = \frac{1} {\sigma \sqrt {2\pi } } exp\{ \frac{ - (y_i - x_iw)^2 }{2 \sigma^2} \}\]

Our task is to estimate \(\hat{w}\) such that the likelihood of \(p\) is maximized. This, in other words, means we need to find the value of \(w\) that maximizes the above expression:

\[\begin{aligned} & \hat{w} = \underset{w}{\text{argmax}} \prod_{i=1}^N p(y_i|x_i; w_i, \sigma^2) \\ \implies & \hat{w} = \underset{w}{\text{argmax}} \prod_{i=1}^N \frac{1} {\sigma \sqrt {2\pi } } exp\{ \frac{ - (y_i - x_iw)^2 }{2 \sigma^2} \} \end{aligned}\]

we can again do the log trick to make this a sum maximization :

\[\begin{aligned} &\hat{w} = \underset{w}{\text{argmax}} \sum_{i=1}^N Log(\frac{1} {\sigma \sqrt {2\pi } } exp\{ \frac{ - (y_i - x_iw)^2 }{2 \sigma^2} \}) \\ \implies &\hat{w} = \underset{w}{\text{argmax}} \{ \sum_{i=1}^N Log(\frac{1}{\sigma \sqrt {2\pi}}) + \sum_{i=1}^N Log(exp\{ \frac{ - (y_i - x_iw)^2 }{2 \sigma^2} \}) \} \\ \implies &\hat{w} = \underset{w}{\text{argmax}} -\sum_{i=1}^N \frac{(y_i - x_iw)^2}{2 \sigma^2} \end{aligned}\]

This is basically the same as the normal expression we had, the only difference being the normalizing factor \(\sigma\). If we change the negative maximization to minimization:

\[\hat{w} = \frac{1}{2 \sigma^2} \underset{w}{\text{argmin}} \sum_{i=1}^N (y_i - x_iw)^2\]

Which is the same as minimizing :

\[\hat{w} = \underset{w}{\text{argmin}} \sum_{i=1}^N (y_i - x_iw)^2\]

and the solution is:

\[\mathbf{w} = (\mathbf{X}^T\mathbf{X})^{-1} \mathbf{X}^T\mathbf{y}\]

Surprise, Surprise!!