Model Uncertainty and Missing Data: An Objective Bayesian Perspective

TL;DR

This paper explores the integration of model uncertainty and missing data within an objective Bayesian framework, proposing methods for variable selection, model comparison, and handling missing data that outperform existing approaches in simulations.

Contribution

It introduces a Bayesian approach that unifies missing data handling and model uncertainty, emphasizing prior predictive marginals and providing a comprehensive methodology for regression and model comparison.

Findings

Method outperforms or matches existing techniques in simulations.

Performance improves with higher missing data percentages.

Approach effectively handles variable selection and model uncertainty.

Abstract

The interplay between missing data and model uncertainty -- two classic statistical problems -- leads to primary questions that we formally address from an objective Bayesian perspective. For the general regression problem, we discuss the probabilistic justification of Rubin's rules applied to the usual components of Bayesian variable selection, arguing that prior predictive marginals should be central to the pursued methodology. In the regression settings, we explore the conditions of prior distributions that make the missing data mechanism ignorable. Moreover, when comparing multiple linear models, we provide a complete methodology for dealing with special cases, such as variable selection or uncertainty regarding model errors. In numerous simulation experiments, we demonstrate that our method outperforms or equals others, in consistently producing results close to those obtained using the full dataset. In general, the difference increases with the percentage of missing data and the correlation between the variables used for imputation. Finally, we summarize possible directions for future research.