The problem at hand is to try to fit the equation $$ \begin{align} y = f(x) + \epsilon, \tag{34} \end{align} $$
where \( f(x) \) is some unknown function of the data \( x \) and \( \epsilon \) is normally distributed with mean zero noise with standard deviation \( \sigma_{\epsilon} \). Our job is to try to find a predictor which estimates the function \( f(x) \). In linear regression we assume that we can formulate the problem as $$ \begin{align} y = X\omega + \epsilon, \tag{35} \end{align} $$
where \( X \) and \( \omega \) are now matrices. Our job at hand is now to find a cost function \( C \), which we wish to minimize in order to find the best estimate of \( \omega \).