Posterior propriety for Bayesian binomial regression models with a parametric family of link functions

We consider a Bayesian analysis of Binomial response data with covariates. To describe the problem under investigation, suppose we have nn independent binomial observations Y1,…,YnY1,…,Yn where Yi∼Bin(mi,θi)Yi∼Bin(mi,θi) and let xi be pp-dimensional covariate vector associated with YiYi for i=1,…,ni=1,…,n. Binomial observations can be analyzed through a generalized linear model (GLM) where we assume θi=F(xiTβ) for some known distribution function F(⋅)F(⋅) and β is the vector of unknown regression coefficients. In this paper, we state necessary and sufficient conditions for propriety of the posterior distribution of β if an improper uniform prior is used on β. We also consider situations where the link function is not pre-specified but belongs to a parametric family and the link function parameters are estimated along with the regression coefficients. In this case, we investigate the propriety of the joint posterior distributions of β and the link function parameters. There are a number of parametric families of link functions available in the literature. As a specific example, we consider Pregibon’s (1980) [17] link function and show that our general posterior propriety results can be used to establish propriety of the posterior distributions corresponding to the Pregibon’s (1980) [17] link. We show that Pregibon’s (1980) [17] simple one parameter family of link function can be used to fit both positively and negatively skewed response curves. Moreover, the conditions for posterior propriety corresponding to the Pregibon’s (1980) [17] link can be easily checked and are milder than those required by the flexible GEV link of Wang and Dey (2010) [24]. As an illustration, we analyze a data set from Ramsey and Schafer (2002) [18] regarding the relationship between dose of Aflatoxicol and odds of liver tumor in rainbow trouts. In this example, the symmetric logit link fails to fit the data, whereas Pregibon’s (1980) [17] skewed link yields a slightly better fit than the GEV link.