Generalized Linear Models

Summary

Chapter 16 of BDA3 covers the Bayesian treatment of GLMs — logistic, Poisson, and other models with non-normal likelihoods. Bayesian priors provide regularization that is especially valuable for separation in logistic regression.

GLM Framework

A GLM has three components:

1. Distribution: $y_i\sim \text{ExponentialFamily}({\eta }_i)$
2. Linear predictor: ${\eta }_i=X_i\beta$
3. Link function: $g({\mu }_i)={\eta }_i$

Model	Distribution	Link
Linear regression	Normal	Identity
Logistic regression	Binomial	Logit
Poisson regression	Poisson	Log

Weakly Informative Priors for Logistic Regression

Tip

For logistic regression, a weakly informative prior like ${\beta }_j\sim t_7(0,2.5)$ (on standardized predictors) prevents separation problems and stabilizes estimates when data are sparse.

Key Applications

• Overdispersed Poisson regression: modeling police stops with extra-Poisson variation
• State-level opinion estimation: multilevel regression with poststratification (MRP)
• Multivariate/multinomial responses: extending to multiple outcome categories
• Loglinear models: for contingency table data

See Also

• Bayesian Linear Regression — the normal special case
• Hierarchical Linear Models — adding group-level structure
• Nonparametric Models Overview — when GLM linearity is too restrictive
• Discrete Choice Models — GLMs with categorical/multinomial likelihood for econometric choice data
• Quantile Regression — Bayesian quantile regression uses the asymmetric Laplace, a GLM-family distribution
• Monsters and Mixtures — zero-inflated and hurdle extensions of Poisson/binomial GLMs for over-dispersed count data
• Hierarchical Models — the natural next step: add group-level structure to any GLM via partial pooling