Jeffreys' prior, finiteness and shrinkage in binomial-response generalized linear models
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This paper studies the finiteness properties of a reduced-bias estimator for logistic regression that results from penalization of the likelihood by Jeffreys' invariant prior; and it provides geometric insights on the shrinkage towards equiprobability that the penalty induces. Some implications of finiteness and shrinkage for inference are discussed, particularly when inference is based on Wald-type procedures. We show how the finiteness and shrinkage properties continue to hold for link functions other than the logistic, and also when the Jeffreys prior penalty is raised to a positive power. In that more general context, we show how maximum penalized likelihood estimates can be obtained by using repeated maximum likelihood fits on iteratively adjusted binomial responses and totals, and we provide a general algorithm for maximum penalized likelihood estimation. These theoretical results and methods underpin the increasingly widespread use of reduced-bias and similarly penalized logistic regression models in many applied fields. We illustrate the results here in one specific context, a Bradley-Terry model to estimate the relative strengths of NBA basketball teams.
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