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

  1. unit information prior

  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

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