The assumption we have made here can be summarized as (and this is going to be useful when we discuss the bias-variance trade off) that there exists a function \( f(\boldsymbol{x}) \) and a normal distributed error \( \boldsymbol{\varepsilon}\sim \mathcal{N}(0, \sigma^2) \) which describe our data
$$ \boldsymbol{y} = f(\boldsymbol{x})+\boldsymbol{\varepsilon} $$We approximate this function with our model from the solution of the linear regression equations, that is our function \( f \) is approximated by \( \boldsymbol{\tilde{y}} \) where we want to minimize \( (\boldsymbol{y}-\boldsymbol{\tilde{y}})^2 \), our MSE, with
$$ \boldsymbol{\tilde{y}} = \boldsymbol{X}\boldsymbol{\beta}. $$