# Augmenting the Cost Function with a Gaussian Prior

I'm doing an online course and one of the assignment says:

To avoid parameters from exploding or becoming highly correlated, it is helpful to augment our cost function with a Gaussian prior: this tends to push parameter weights closer to zero, without constraining their direction, and often leads to classifiers with better generalization ability.

Exactly what does "augment our cost function with a Gaussian prior" mean, and how could I go about doing that?

Let's look at the example of linear regression. Instead of deriving it from solving the normal equations, we can motivate it by thinking about it as finding the conditional distribution $P(y|x)$. Let's assume that this distribution follows a Gaussian distribution with fixed variance $\sigma^2$ and mean $\hat{y}(x,w)$ and weights $w$. Assuming that samples are i.i.d. we can easily show that applying maximum-likelihood gives the same values of $w$ than minimizing the mean squared error. See example 5.5.1 for a mathematical explanation.
If we additionally assume a prior Gaussian distribution for $P(w)$ (w.l.o.g. we can assume unit variance $I$), we can now show that the posterior $P(w | x,y)$ of $w$ is also Gaussian distributed (that is because the prior is "conjugate") with variance $\frac{1}{\alpha} I$ and we recover linear regression with weight decay $\alpha w^T w=\alpha \lVert w \rVert^2_2$ (i.e. $L^2$ or Tikhonov regularization).