# Jeffreys-prior penalty, finiteness and shrinkage in binomial-response generalized linear models

## TL;DR

This paper demonstrates that Jeffreys' prior penalty ensures finite estimates in binomial GLMs, reduces bias, and induces shrinkage, with practical computation methods and implications for inference.

## Contribution

It establishes the finiteness and shrinkage properties of Jeffreys-prior penalized binomial models and develops a practical iterative computation procedure.

## Key findings

- Penalization yields finite maximum likelihood estimates.
- Jeffreys-prior reduces asymptotic bias in logistic regression.
- Shrinkage towards equiprobability is theoretically confirmed.

## Abstract

Penalization of the likelihood by Jeffreys' invariant prior, or by a positive power thereof, is shown to produce finite-valued maximum penalized likelihood estimates in a broad class of binomial generalized linear models. The class of models includes logistic regression, where the Jeffreys-prior penalty is known additionally to reduce the asymptotic bias of the maximum likelihood estimator; and also models with other commonly used link functions such as probit and log-log. Shrinkage towards equiprobability across observations, relative to the maximum likelihood estimator, is established theoretically and is studied through illustrative examples. Some implications of finiteness and shrinkage for inference are discussed, particularly when inference is based on Wald-type procedures. A widely applicable procedure is developed for computation of maximum penalized likelihood estimates, by using repeated maximum likelihood fits with iteratively adjusted binomial responses and totals. These theoretical results and methods underpin the increasingly widespread use of reduced-bias and similarly penalized binomial regression models in many applied fields.