# Is there a Bayesian interpretation of linear regression with simultaneous L1 and L2 regularization (aka elastic net)?

It's well known that linear regression with an $l^2$ penalty is equivalent to finding the MAP estimate given a Gaussian prior on the coefficients. Similarly, using an $l^1$ penalty is equivalent to using a Laplace distribution as the prior.

It's not uncommon to use some weighted combination of $l^1$ and $l^2$ regularization. Can we say that this is equivalent to some prior distribution over the coefficients (intuitively, it seems that it must be)? Can we give this distribution a nice analytic form (maybe a mixture of Gaussian and Laplacian)? If not, why not?

• see this paper: tandfonline.com/doi/abs/10.1198/jasa.2011.tm09241 (If this isn't properly answered in a week or two, I'll post (more or less) a summary of that paper) Jun 2 '17 at 17:10
• I should add that any time frequentists have a penalty $pen$, a bayesian can interpret that as a (possibly improper) prior $e^{-pen}$ under a standard gaussian model. Jun 2 '17 at 17:12
• thanks, this paper and its citations answer my question perfectly! Jun 2 '17 at 17:15
• Zou and Hastie 2005, which it looks like is the paper introducing the elastic net. They give an interpretation in terms of a prior. Jun 2 '17 at 17:25
• Okay, cool! I think their bayesian interpretation ties into my second comment Jun 2 '17 at 18:22

A Bayesian elastic net representation was proposed by Kyung et. al. in their Section 3.1. Although the prior for the regression coefficient $\beta$ was correct, the authors incorrectly wrote down the mixture representation.