  Selecting important spatial-dependent variables under the nonhomogeneous
spatial Poisson process model is an important topic of great current interest.
In this paper, we use the Deviance Information Criterion (DIC) and Logarithm of
the Pseudo Marginal Likelihood (LPML) for Bayesian variable selection under the
nonhomogeneous spatial Poisson process model. We further derive the new Monte
Carlo estimation formula for LPML in the spatial Poisson process setting.
Extensive simulation studies are carried out to evaluate the empirical
performance of the proposed criteria. The proposed methodology is further
applied to the analysis of two large data sets, the Earthquake Hazards Program
of United States Geological Survey (USGS) earthquake data and the Forest of
Barro Colorado Island (BCI) data.
