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
Then we can construct the following Gibbs sampler:
Weakly informative prior distributions
unit information prior
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
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