  Rapid advancements in spatial technologies including Geographic Information
Systems (GIS) and remote sensing have generated massive amounts of spatially
referenced data in a variety of scientific and data-driven industrial
applications. These advancements have led to a substantial, and still
expanding, literature on the modeling and analysis of spatially oriented big
data. In particular, Bayesian inferences for high-dimensional spatial processes
are being sought in a variety of remote-sensing applications including, but not
limited to, modeling next generation Light Detection and Ranging (LiDAR)
systems and other remotely sensed data. Massively scalable spatial processes,
in particular Gaussian processes (GPs), are being explored extensively for the
increasingly encountered big data settings. Recent developments include GPs
constructed from sparse Directed Acyclic Graphs (DAGs) with a limited number of
neighbors (parents) to characterize dependence across the spatial domain. The
DAG can be used to devise fast algorithms for posterior sampling of the latent
process, but these may exhibit pathological behavior in estimating covariance
parameters. While these issues are mitigated by considering marginalized
samplers that exploit the underlying sparse precision matrix, these algorithms
are slower, less flexible, and oblivious of structure in the data. The current
article introduces the Grid-Parametrize-Split (GriPS) approach for conducting
Bayesian inference in spatially oriented big data settings by a combination of
careful model construction and algorithm design to effectuate substantial
improvements in MCMC convergence. We demonstrate the effectiveness of our
proposed methods through simulation experiments and subsequently undertake the
modeling of LiDAR outcomes and production of their predictive maps using G-LiHT
and other remotely sensed variables.
