Invited Discussion of "Model Uncertainty and Missing Data: An Objective Bayesian Perspective" by Gonzalo Garc\'ia-Donato , Mar\'ia Eugenia Castellanos , Stefano Cabras Alicia Quir\'os , and Anabel Forte

TL;DR

This paper discusses a Bayesian approach to handle model uncertainty and missing data in Gaussian regression, introducing an imputation $g$-prior that simplifies computations and unifies different Bayesian methods.

Contribution

It proposes a novel imputation $g$-prior method that addresses both missing data and model uncertainty simultaneously within an objective Bayesian framework.

Findings

Develops a coherent approach for missing data and model uncertainty

Connects imputation $g$-prior to $g$-prior with imputed $X$

Provides computationally tractable Bayesian model selection

Abstract

The article by Garc{\'i}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 $X_\gamma^TX_\gamma$ for model $\gamma$ in the covariance of $\beta_\gamma$ with $\Sigma_{X_\gamma}$, 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.