I don't know what your experience with Bayesian modeling is, and I'll admit mine is limited, but STAN can solve linear regression by defining up a linear model

I don't know what your experience with Bayesian modeling is, and I'll admit mine is limited, but STAN can solve linear regression by defining up a linear model and then setting the dimension parameters as a normal distribution to solve. This is great because it gives you a measure of certainty for each of your parameters.