Identifiability, improper priors, and Gibbs sampling for generalized linear models

Markov chain Monte Carlo algorithms are widely used in the fitting of gener alized linear models (GLMs). Such model fitting is somewhat of an art form, requiring suitable trickery and tuning to obtain results in which one can have confidence. A wide range of practical issues arise. The focus here is on parameter identifiability and posterior propriety. In particular, we cla rify that nonidentifiability arises for usual GLMs and discuss its implicat ions for simulation-based model fitting. Because often some part of the pri or specification is vague, we consider whether the resulting posterior is p roper, providing rather general and easily checked results for GLMs. We als o show that if a Gibbs sampler is run with an improper posterior, then it m ay be possible to use the output to obtain meaningful inference for certain model unknowns.