Modelling and predicting distribution-valued fields with applications to inversion under uncertainty

Capturing the dependence between a random response and predictors is a fundamental task in statistics and stochastic modelling. The focus of this work is on density regression, which entails estimating response distributions given predictor values. It enables the derivation of various statistical quantities, including the conditional mean, threshold exceedance probabilities, and quantiles. This thesis presents a flexible approach, based upon the class of so-called Spatial Logistic Gaussian Processes (SLGPs). The SLGP framework utilizes a well-behaved latent Gaussian Process that undergoes a non-linear transformation, resulting in a class of models suitable for capturing spatially-dependent probability measures. SLGP models overcome limitations associated with strong distributional assumptions (e.g. shapes constraints, log-concavity, Gaussianity, etc.), varying sample sizes, and changes in target density shapes and modalities. The first part of this work is dedicated to the development of SLGP models and gaining a deep understanding of the associated mathematical concepts. We introduce SLGPs from the perspective of random measures and their densities, and investigate links between properties of SLGPs and underlying processes. We show that SLGP models can be characterized by their log-increments and leverage this characterization to establish theoretical results with a main focus on spatial regularity. We then focus on applicability of our approach, and propose an implementation relying on finite rank Gaussian Processes. We demonstrate it on synthetic examples and on temperature distributions at meteorological stations. Finally, we address the potential of SLGPs for statistical inference, focusing on their potential in stochastic optimization and stochastic inverse problems. Notably, for inverse problems, an Approximate Bayesian Computation (ABC) framework is introduced, leveraging SLGP-surrogated likelihoods to accommodate situations with limited to moderate data. This methodology, inspired by GP-ABC methods, harnesses the probabilistic nature of SLGPs to guide data acquisition, thereby facilitating accelerated inference. We illustrate these approaches on synthetic examples as well as on a hydrogeological inverse problem in which a contaminant source is sought under uncertain geological scenario

Goal-oriented adaptive sampling under random field modelling of response probability distributions

In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision space. We consider cases where the spatial variation of these response distributions does not only concern their mean and/or variance but also other features including for instance shape or uni-modality versus multi-modality. Our contributions build upon a non-parametric Bayesian approach to modelling the thereby induced fields of probability distributions, and in particular to a spatial extension of the logistic Gaussian model. The considered models deliver probabilistic predictions of response distributions at candidate points, allowing for instance to perform (approximate) posterior simulations of probability density functions, to jointly predict multiple moments and other functionals of target distributions, as well as to quantify the impact of collecting new samples on the state of knowledge of the distribution field of interest. In particular, we introduce adaptive sampling strategies leveraging the potential of the considered random distribution field models to guide system evaluations in a goal-oriented way, with a view towards parsimoniously addressing calibration and related problems from non-linear (stochastic) inversion and global optimisation