This model is similar to ridge regression but does not require cross-validation. Works well in case of multicollinearity. You can find python implementation here Matlab implementation here. There is also small tutorial that compares OLS and Bayesian Regression here.

ARD regression is almost identical to Bayesian Regression, the only difference is prior on parameters. Prior used in ARD results in sparse solutions, this can be considered bayesian analog of lasso regression.

Kernelised version of ARD is called Relevance Vector Machine (RVM), it is mainly used for nonlinear regression. In many cases RVM produces solutions comparable to SVM with only a fraction of support vectors used in SVM. Python code for RVM and ARD can be found here and demo is here