Hi Chris,
We already talked about that in this post.
In a nutshell, you won’t have a one-to-one equivalence in the Bayesian framework – just putting sensible priors is a form of regularization. So I think it’s not a binary situation in the Bayesian framework – either you regularize or you don’t. It’s more like a knob that you adjust – you penalize parameters more or less, depending on your priors and use case.
Hope this helps 
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