Bayesian interpretation of ridge

With Gaussian observation errors and an isotropic zero-mean Gaussian coefficient prior, explain the connection between MAP estimation and ridge.

The negative log likelihood is proportional to squared residual error divided by the error variance. The negative log prior is proportional to the squared coefficient norm divided by the prior variance. Their sum is a ridge objective, with penalty strength equal to error variance divided by prior variance when the loss is an unnormalised residual sum of squares. Different averaging conventions change this numerical relationship.