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The equivalence of a LASSO, or Ridge, regression can be obtained in the Bayesian framework by using a specific prior combined with the posterior mode estimate (e.g., https://osf.io/cg8fq/). What is the equivalence of the LASSO trace plot in the Bayesian framework? And how can it be obtained in Stan?
An exemplary trace plot is given below:
It can be argued that the posterior mode estimate is not really Bayesian. When you integrate over the posterior, their effect for predictive mean is very unlikely to be exactly 0. See also The king must die | Statistical Modeling, Causal Inference, and Social Science
The question is not meaningful if we consider that combining Bayesian inference and Laplace prior is not a good idea. Also if we would combine them, we would like to integrate over the unknown prior term.
In theory, you could use Stan for the original LASSO, but not as efficiently as it would be possible.
Now after providing these negative answers, I’d like to ask what is behind your questions? Is there something you would like to achieve? If you are interested in variable selection in Bayesian context, you might look at Model selection tutorials and talks. These tutorials discuss projection predictive variable selection which can order the variables and show how close you can get the full model predictive performance after including k variables. Would that be useful for you?
Thanks for your helpful answer!
In Erb et al. (2019) the idea is described as “using a specific prior combined with the posterior mode estimate”. They also discuss Laplace priors (3.3). Further, Laplace priors for regularization are also discussed here: Regularized bayesian logistic regression in JAGS - Cross Validated
Yes, this is what we are working on. We have an empty hierarchical Bayesian model and want to test a set of covariates and know how much predictive information each covariate contains/how relevant it is for predicting the outcome.