  To overcome challenges in fitting complex models with small samples,
catalytic priors have recently been proposed to stabilize the inference by
supplementing observed data with synthetic data generated from simpler models.
Based on a catalytic prior, the Maximum A Posteriori (MAP) estimator is a
regularized estimator that maximizes the weighted likelihood of the combined
data. This estimator is straightforward to compute, and its numerical
performance is superior or comparable to other likelihood-based estimators. In
this paper, we study several theoretical aspects regarding the MAP estimator in
generalized linear models, with a particular focus on logistic regression. We
first prove that under mild conditions, the MAP estimator exists and is stable
against the randomness in synthetic data. We then establish the consistency of
the MAP estimator when the dimension of covariates diverges slower than the
sample size. Furthermore, we utilize the convex Gaussian min-max theorem to
characterize the asymptotic behavior of the MAP estimator as the dimension
grows linearly with the sample size. These theoretical results clarify the role
of the tuning parameters in a catalytic prior, and provide insights in
practical applications. We provide numerical studies to confirm the effective
approximation of our asymptotic theory in finite samples and to illustrate
adjusting inference based on the theory.
