Generalized linear models have become a standard class of models for data analysts. However, in some applications, heterogeneity in samples is too great to be explained by the simple variance function implicit in such models. Utilizing a two parameter exponential family which is overdispersed relative to a specified one-parameter exponential family enables the creation of classes of overdispersed generalized linear models (OGLMs) which are analytically attractive. We propose fitting such models within a Bayesian framework employing noninformative priors in order to let the data drive the inference. Hence, our analysis approximates likelihood-based inference but with possibly more reliable estimates of variability for small sample sizes. Bayesian calculations are carried out using a Metropolis-within-Gibbs sampling algorithm. An illustrative example using a data set involving damage incidents to cargo ships is presented. Details of the data analysis are provided including comparison with the standard generalized linear models analysis. Several diagnostic tools reveal the improved performance of the OGLM. © 1997 Elsevier Science B.V.