Open Access
December 2008 A weakly informative default prior distribution for logistic and other regression models
Andrew Gelman, Aleks Jakulin, Maria Grazia Pittau, Yu-Sung Su
Ann. Appl. Stat. 2(4): 1360-1383 (December 2008). DOI: 10.1214/08-AOAS191


We propose a new prior distribution for classical (nonhierarchical) logistic regression models, constructed by first scaling all nonbinary variables to have mean 0 and standard deviation 0.5, and then placing independent Student-t prior distributions on the coefficients. As a default choice, we recommend the Cauchy distribution with center 0 and scale 2.5, which in the simplest setting is a longer-tailed version of the distribution attained by assuming one-half additional success and one-half additional failure in a logistic regression. Cross-validation on a corpus of datasets shows the Cauchy class of prior distributions to outperform existing implementations of Gaussian and Laplace priors.

We recommend this prior distribution as a default choice for routine applied use. It has the advantage of always giving answers, even when there is complete separation in logistic regression (a common problem, even when the sample size is large and the number of predictors is small), and also automatically applying more shrinkage to higher-order interactions. This can be useful in routine data analysis as well as in automated procedures such as chained equations for missing-data imputation.

We implement a procedure to fit generalized linear models in R with the Student-t prior distribution by incorporating an approximate EM algorithm into the usual iteratively weighted least squares. We illustrate with several applications, including a series of logistic regressions predicting voting preferences, a small bioassay experiment, and an imputation model for a public health data set.


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Andrew Gelman. Aleks Jakulin. Maria Grazia Pittau. Yu-Sung Su. "A weakly informative default prior distribution for logistic and other regression models." Ann. Appl. Stat. 2 (4) 1360 - 1383, December 2008.


Published: December 2008
First available in Project Euclid: 8 January 2009

zbMATH: 1156.62017
MathSciNet: MR2655663
Digital Object Identifier: 10.1214/08-AOAS191

Keywords: Bayesian inference , generalized linear model , hierarchical model , least squares , Linear regression , logistic regression , multilevel model , noninformative prior distribution , weakly informative prior distribution

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.2 • No. 4 • December 2008
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