# Robust regression with M-estimators

I have a couple of question regarding robust regression with M-estimators, such as Huber estimator or Tukey biweight estimator:

1. Is it possible/common to combine these with regularization terms, such as those used in ridge and lasso regression?
2. Given that M-estimators originate from maximum likelihood, do they occur in bayesian framework? (and how)

Both extensions seem obvious to me, but searching the internet for either produced surprisingly few hits, so I expect that there are some caveats.

2. While M-estimation arises from likelihood ideas, it doesn't always correspond to maximizing a likelihood. Some M-estimators do. The Huber does, for example. However, some don't - like the Tukey biweight (indeed any M-estimator whose $$\psi$$-function cuts off/goes to 0 at some finite value can't be MLE for any density). The psi function can redescend and you may still have it correspond to likelihood (e.g. it does with the Cauchy) but for the $$\psi$$-function to correspond to a density, it can't descend too quickly -- or the corresponding density would not be integrable.