Bayesian geostatistical modeling with informative sampling locations

We consider geostatistical models that allow the locations at which data are collected to be informative about the outcomes. Diggle et al. [2009] refer to this problem as preferential sampling, though we use the term informative sampling to highlight the relationship with the longitudinal data literature on informative observation times. In the longitudinal setting, joint models of the observation times and outcome process are widely used to adjust for informative sampling bias. We propose a Bayesian geostatistical joint model, which models the locations using a log Gaussian Cox process, while modeling the outcomes conditionally on the locations as Gaussian with a Gaussian process spatial random effect and adjustment for the location intensity process. We prove posterior propriety under an improper prior on the parameter controlling the degree of informative sampling, demonstrating that the data are informative. In addition, we show that the density of the locations and mean function of the outcome process can be estimated consistently under mild assumptions. The methods are applied to ozone data.