A default prior for regression coefficients

TL;DR

This paper proposes a new default Bayesian prior for regression coefficients, suggesting a normal distribution with mean zero and standard deviation equal to the estimator's standard error, which better aligns with practical inference needs.

Contribution

It introduces a more suitable default prior for regression coefficients, replacing the common uniform prior, based on theoretical and empirical justifications.

Findings

The proposed prior is non-informative for the sign of the coefficient.

It aligns well with meta-analysis data from MEDLINE.

Outperforms the uniform prior in practical inference scenarios.

Abstract

When the sample size is not too small, M-estimators of regression coefficients are approximately normal and unbiased. This leads to the familiar frequentist inference in terms of normality-based confidence intervals and p-values. From a Bayesian perspective, use of the (improper) uniform prior yields matching results in the sense that posterior quantiles agree with one-sided confidence bounds. For this, and various other reasons, the uniform prior is often considered objective or non-informative. In spite of this, we argue that the uniform prior is not suitable as a default prior for inference about a regression coefficient in the context of the bio-medical and social sciences. We propose that a more suitable default choice is the normal distribution with mean zero and standard deviation equal to the standard error of the M-estimator. We base this recommendation on two arguments. First, we show that this prior is non-informative for inference about the sign of the regression coefficient. Secondly, we show that this prior agrees well with a meta-analysis of 50 articles from the MEDLINE database.