I'm working on a Bayesian Regression problem where I would like to estimate the beta coefficients subject to this constraint (penalty):
$\sum|\beta_i|<C$ or similarly $\sum \beta_i^2<C$
Which is basically a Lasso or L2 Penalty.
Now, if I understand correctly, we constrain the coefficients through the prior in Bayesian analysis. Therefore my question is what would an appropriate prior be for the Betas? I should note, that for my case, the betas are restricted to be positive, they cannot be negative.