Compatibility of Prior Specifications Across Linear Models

TL;DR

This paper explores methods for deriving compatible priors across nested linear models to improve Bayesian model comparison, focusing on techniques like marginalization, conditioning, and Kullback-Leibler projection, with practical illustrations and evaluations.

Contribution

It reviews and compares various procedures for deriving priors in submodels from a full model, enhancing prior compatibility and robustness in Bayesian linear model selection.

Findings

Kullback-Leibler projection offers a flexible way to derive priors.

Methods are evaluated through simulations and real data analysis.

Compatible priors improve model comparison reliability.

Abstract

Bayesian model comparison requires the specification of a prior distribution on the parameter space of each candidate model. In this connection two concerns arise: on the one hand the elicitation task rapidly becomes prohibitive as the number of models increases; on the other hand numerous prior specifications can only exacerbate the well-known sensitivity to prior assignments, thus producing less dependable conclusions. Within the subjective framework, both difficulties can be counteracted by linking priors across models in order to achieve simplification and compatibility; we discuss links with related objective approaches. Given an encompassing, or full, model together with a prior on its parameter space, we review and summarize a few procedures for deriving priors under a submodel, namely marginalization, conditioning, and Kullback--Leibler projection. These techniques are illustrated and discussed with reference to variable selection in linear models adopting a conventional $g$-prior; comparisons with existing standard approaches are provided. Finally, the relative merits of each procedure are evaluated through simulated and real data sets.