  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.
