Model Selection for Semiparametric Bayesian Models with Application to Overdispersion

In analyzing complicated data, we are often unwilling or not confident to impose a parametric model for the data-generating structure. One important example is data analysis for proportional or count data with overdispersion. The obvious advantage of assuming full parametric models is that one can resort to likelihood analyses, for instance, to use AIC or BIC to choose the most appropriate regression models. For overdispersed proportional data, possible parametric models include the Beta-binomial models, the double exponential models, etc. In this paper, we extend the generalized linear models by replacing the full parametric models with a finite number of moment restrictions on both the data and the structural parameters. For such semiparametric Bayesian models, we propose a method for selecting the best possible regression model in the semiparametric model class. We will apply the proposed model selection technique to overdispersed data. We will demonstrate the use of the proposed semiparametric information criterion using the well-known data on germination of Orobanche.