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?