Properties and Implementation of Jeffreys's Prior in Binomial Regression Models.

Area of Science:

• Statistics
• Statistical Modeling
• Bayesian Inference

Background:

• Jeffreys' prior is a non-informative prior distribution used in Bayesian statistics.
• Binomial regression models are widely used for analyzing binary outcome data.
• Understanding the theoretical properties of priors is crucial for robust Bayesian analysis.

Purpose of the Study:

• To investigate the theoretical properties of Jeffreys' prior specifically for binomial regression models.
• To characterize the tail behavior and invariance properties of Jeffreys' prior in this context.
• To establish connections with model selection criteria and develop computational methods.

Main Methods:

• Theoretical analysis of Jeffreys' prior for binomial regression with logistic, probit, and log-log links.
• Comparison with multivariate t and normal distributions for tail behavior characterization.
• Development of an importance sampling algorithm for prior and posterior computations.

Main Results:

• Jeffreys' prior is shown to be symmetric and unimodal for a class of binomial regression models.
• Prior and posterior normalizing constants are invariant to linear covariate transformations.
• A theoretical link between the Bayes Information Criterion and induced dimension penalty is established.

Conclusions:

• Jeffreys' prior exhibits desirable theoretical properties for binomial regression models.
• The established connections and computational methods facilitate its application in variable selection.
• The proposed methods are illustrated with a real data analysis.