Is Bayesian Ridge Regression another name of Bayesian Linear Regression?

Question

I searched about Bayesian Ridge Regression on Internet but most of the result i became is about Bayesian Linear Regression. I wonder if it's both the same things because the formula look quite similar

Answer 1

Ridge regression uses regularization with $L_2$ norm, while Bayesian regression, is a regression model defined in probabilistic terms, with explicit priors on the parameters. The choice of priors can have the regularizing effect, e.g. using Laplace priors for coefficients is equivalent to $L_1$ regularization. They are not the same, because ridge regression is a kind of regression model, and Bayesian approach is a general way of defining and estimating statistical models that can be applied to different models.

For reading more on ridge regression and regularization see the threads: Why does ridge estimate become better than OLS by adding a constant to the diagonal?, and What problem do shrinkage methods solve?, and When should I use lasso vs ridge?, and Why is ridge regression called "ridge", why is it needed, and what happens when $\lambda$ goes to infinity?, and many others we have.