# Bayesian Model Selection for Misspecified Models in Linear Regression

## TL;DR

This paper develops a unified Bayesian approach that combines the strengths of BIC and AIC for linear regression, enhancing robustness against model misspecification and low signal-to-noise scenarios.

## Contribution

It introduces a novel prior in an augmented model-plus-noise space that unifies BIC and AIC assumptions, improving model selection under misspecification.

## Key findings

- Unified prior inherits properties of BIC and AIC
- Enhanced robustness to model misspecification
- Applicable in low signal-to-noise ratio conditions

## Abstract

While the Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) are powerful tools for model selection in linear regression, they are built on different prior assumptions and thereby apply to different data generation scenarios. We show that in the finite-dimensional case their respective assumptions can be unified within an augmented model-plus-noise space and construct a prior in this space which inherits the beneficial properties of both AIC and BIC. This allows us to adapt the BIC to be robust against misspecified models where the signal to noise ratio is low.