Linear Regression
The variable selection problem has a natural Bayesian solution: Any collection of models having different sets of regressors can be computed via their Bayes factors.

# A semiconjugate prior distribution

Let
$\beta \sim MN(\boldsymbol\beta_0, \Sigma_0)$
, then
$\beta\mid \mathbf y,\mathbf X, \sigma^2 \sim N((\Sigma_0^{-1}+\mathbf X^T\mathbf X/\sigma^2)^{-1}(\Sigma_0^{-1}\boldsymbol\beta_0+\mathbf X^T\mathbf y/\sigma^2), (\Sigma_0^{-1}+\mathbf X^T\mathbf X/\sigma^2)^{-1})$
and let
$1/\sigma^2\sim Ga(\nu_0/2,\nu_0\sigma_0^2/2)$
, then
$1/\sigma^2\mid \mathbf y,\mathbf X,\boldsymbol\beta \sim Ga([\nu_0+n]/2, [\nu_0\sigma_0^2+SSR(\boldsymbol \beta)]/2)$
Then we can construct the following Gibbs sampler:

# Weakly informative prior distributions

1. 1.
unit information prior
2. 2.
the parameter estimation should be invariant to changes in the scale of the regressors.
For the second case, we can derive a Monte Carlo approximation:
since
$p(\sigma^2,\boldsymbol\beta\mid \mathbf y, \mathbf X)\propto p(\boldsymbol\beta\mid \mathbf y,\mathbf X,\sigma^2)p(\sigma^2\mid \mathbf y,\mathbf X)$