Bayesian linear model vs classical linear model

Question

My question is a quite general and simple question.

I was wondering what is the advantage of a bayesian linear model over a classical linear model if, for the bayesian model, only non-informative priors are used?

Shouldn't be more or less the same?

What I mean by linear model is the well-known model that links an observation vector of dependent variables $\mathbf{Y}$ of size $n$ and a matrix $\mathbf{X}$ (design matrix) containing the information about independent variable of size $n\times p$, $p\leq n$, such that $$ \mathbf{Y} = \mathbf{X}\theta, $$ where $\theta$ is the parameter to be optimized.

Answer 1

Bayesian linear regression is the same as linear regression. If you find the parameters of linear regression with maximum likelihood, the difference in the case of a Bayesian flavor is that it would also consider the priors. You are asking about uninformative priors, but there is no such a thing as "uninformative" prior. Every prior does affect the result, some just do this only slightly. The exception would be if you used constant, improper prior $p(\theta) \propto 1$, that would not affect the result, and the Bayesian maximum a posteriori estimate would be equal to the maximum likelihood result.

Your second question about the advantages of using a Bayesian model with uninformative priors is answered in the Why would someone use a Bayesian approach with a 'noninformative' improper prior instead of the classical approach? thread.

Answer 2

You get the Bayesian interpretation of your parameters + any other benefits/features of a Bayesian approach. Numerically, everything will be the same (or very similar)