Bayesian proxy to Huber-White Estimators


Less is more. Stay pure. Stay poor.
I am working on a study where a very small portion of the sample contribute more than one observation. I was wondering if there was a Bayesian proxy to the Huber-White estimator or robust regression? I was actually planning on using non-informative priors since it is novel research and there will be multiple variables in the model. Non-informative priors would just let the estimates be the estimates. I know you can use priors to regularize/shrink estimates, but I guess I am looking to the opposite, broadening the certainty of the estimates. I have seen a little in Stan on this, but I was wonder if anyone could think of a basic simple hack to address this via the selection of priors.