  Double generalized linear models provide a flexible framework for modeling
data by allowing the mean and the dispersion to vary across observations.
Common members of the exponential dispersion family including the Gaussian,
Poisson, compound Poisson-gamma (CP-g), Gamma and inverse-Gaussian are known to
admit such models. The lack of their use can be attributed to ambiguities that
exist in model specification under a large number of covariates and
complications that arise when data display complex spatial dependence. In this
work we consider a hierarchical specification for the CP-g model with a spatial
random effect. The spatial effect is targeted at performing uncertainty
quantification by modeling dependence within the data arising from location
based indexing of the response. We focus on a Gaussian process specification
for the spatial effect. Simultaneously, we tackle the problem of model
specification for such models using Bayesian variable selection. It is effected
through a continuous spike and slab prior on the model parameters, specifically
the fixed effects. The novelty of our contribution lies in the Bayesian
frameworks developed for such models. We perform various synthetic experiments
to showcase the accuracy of our frameworks. They are then applied to analyze
automobile insurance premiums in Connecticut, for the year of 2008.
