Power-Conditional-Expected Priors: Using <i>g</i>-Priors With Random Imaginary Data for Variable Selection

The Zellner's <i>g</i>-prior and its recent hierarchical extensions are the most popular default prior choices in the Bayesian variable selection context. These prior setups can be expressed as power-priors with fixed set of imaginary data. In this article, we borrow ideas from the power-expected-posterior (PEP) priors to introduce, under the <i>g</i>-prior approach, an extra hierarchical level that accounts for the imaginary data uncertainty. For normal regression variable selection problems, the resulting power-conditional-expected-posterior (PCEP) prior is a conjugate normal-inverse gamma prior that provides a consistent variable selection procedure and gives support to more parsimonious models than the ones supported using the <i>g</i>-prior and the hyper-<i>g</i> prior for finite samples. Detailed illustrations and comparisons of the variable selection procedures using the proposed method, the <i>g</i>-prior, and the hyper-<i>g</i> prior are provided using both simulated and real data examples. Supplementary materials for this article are available online.