  We investigate a class of models for non-parametric estimation of probability
density fields based on scattered samples of heterogeneous sizes. The
considered SLGP models are Spatial extensions of Logistic Gaussian Process
models and inherit some of their theoretical properties but also of their
computational challenges. We introduce SLGPs from the perspective of random
measures and their densities, and investigate links between properties of SLGPs
and underlying processes. Our inquiries are motivated by SLGP's abilities to
deliver probabilistic predictions of conditional distributions at candidate
points, to allow (approximate) conditional simulations of probability
densities, and to jointly predict multiple functionals of target distributions.
We show that SLGP models induced by continuous GPs can be characterized by the
joint Gaussianity of their log-increments and leverage this characterization to
establish theoretical results pertaining to spatial regularity. We extend the
notion of mean-square continuity to random measure fields and establish
sufficient conditions on covariance kernels underlying SLGPs for associated
models to enjoy such regularity properties. From the practical side, we propose
an implementation relying on Random Fourier Features and demonstrate its
applicability on synthetic examples and on temperature distributions at
meteorological stations, including probabilistic predictions of densities at
left-out stations.
