Invited Discussion of "Model Uncertainty and Missing Data: An Objective Bayesian Perspective" by Gonzalo García-Donato , María Eugenia Castellanos , Stefano Cabras Alicia Quirós , and Anabel Forte

The article by Garc{í}a-Donato and co-authors addresses the dual challenges of accounting for model uncertainty and missing data within the Gaussian regression frameworks from an objective Bayesian perspective. Thru the use of an imputation $g$-prior that replaces $X_{\gamma }^T{X}_{\gamma }$ for model $\gamma$ in the covariance of ${\beta }_{\gamma }$ with ${\mathrm{\Sigma }}_{X_{\gamma }}$, the authors develop a coherent approach to addressing the missing data problem and model uncertainty simultaneously with random $X_{\gamma }$ in the missing at random (MAR) or missing completely at random (MCAR) settings, while still being computationally tractable. I discuss the connection of the imputation $g$-prior to the $g$-prior with imputed $X$, and to model selection for graphical models that provide an alternative justification for the $g$-prior for random $X$s.