  Logistic regression models for binomial responses are routinely used in
statistical practice. However, the maximum likelihood estimate may not exist
due to data separability. We address this issue by considering a conjugate
prior penalty which always produces finite estimates. Such a specification has
a clear Bayesian interpretation and enjoys several invariance properties,
making it an appealing prior choice. We show that the proposed method leads to
an accurate approximation of the reduced-bias approach of Firth (1993),
resulting in estimators with smaller asymptotic bias than the
maximum-likelihood and whose existence is always guaranteed. Moreover, the
considered penalized likelihood can be expressed as a genuine likelihood, in
which the original data are replaced with a collection of pseudo-counts. Hence,
our approach may leverage well established and scalable algorithms for logistic
regression. We compare our estimator with alternative reduced-bias methods,
vastly improving their computational performance and achieving appealing
inferential results.
