Abstract

Spatial Logistic Gaussian Processes (SLGPs) offer a probabilistic framework for modelling random fields of probability measures indexed by arbitrary covariates. Constructed by applying a non-linear transformation to a latent Gaussian Process, SLG Ps are models flexible enough to encode complex variations in shape and modality. Crucially, they enable probabilistic inference in settings with scattered, irregularly spaced, or non-replicated data. This talk explores both theoretical and practical contributions. On the theoretical side, we formalise SLG Ps as representers of random probability measure fields and investigate their structure and smoothness properties. We extend classical spatial statistics notions such as expected mean-square continuity to the distributional setting, and we establish sufficient conditions on the covariance kernel of the underlying GP to ensure spatial regularity with respect to statistical dissimilarities between distributions (e.g., Hellinger, KL, total variation).From a practical perspective, we introduce an implementation based on finite-rank approximations using Random Fourier Features, and propose several inference schemes—MCMC, MAP , and Laplace approximation—each balancing statistical fidelity and computational cost. We illustrate the flexibility and expressiveness of the SLGP models through experiments on synthetic and real-world datasets, including challenging non-replicated and heterogeneously sampled settings.