Ensemble-based methods are powerful tools for Bayesian inference in complex, high-dimensional systems, combining computational efficiency with accuracy. Originating in data assimilation, the ensemble Kalman filter now serves as a standard technique for inverse problems across disciplines, including weather forecasting and signal processing. Building on this foundation, this thesis advances ensemble transform methods along three key directions: (i) extending beyond the quadratic loss, (ii) formulating affine-invariant techniques, and (iii) employing gradient-free approximations. Together, these contributions provide a robust framework for Bayesian inference in modern machine learning, generative modeling, and cognitive science. In the first part, we extend ensemble transform methods beyond the quadratic loss by developing affine-invariant formulations for classification tasks. Through numerical experiments with Bayesian neural networks, we show how these methods provide improved predictive uncertainty estimates compared to state-of-the-art techniques, reducing the model's overconfidence on out-of-distribution data. The second part of this work examines the novel application of ensemble Kalman–based methods within cognitive science, with a particular emphasis on models of natural language comprehension. While uncertainty representation is central to human cognition, it is largely absent from standard neural network models. By framing sentence comprehension as a Bayesian inverse problem, we extend ensemble Kalman methods to equip the Sentence Gestalt model with predictive uncertainty. This enhancement enables the model to capture human-like patterns of uncertainty when processing ambiguous sentences, thereby narrowing the gap between deterministic neural architectures and probabilistic theories of language understanding. Finally, the third part introduces a gradient-free ensemble transform Langevin dynamics method for likelihood-free inference using maximum mean discrepancy. This framework enables generalized Bayesian inference in simulator-based models without requiring explicit likelihoods and simulator gradients. We further demonstrate its robustness to model misspecification in generative modeling, and effectiveness in chaotic dynamical systems. The novel methods and applications developed in this thesis extend ensemble transform techniques to address predictive reliability in modern machine learning and simulator-based systems. At the same time, they build a bridge between methodological innovation and cross-disciplinary application, showing that the proposed techniques can enhance uncertainty-aware modeling in complex artificial systems and deepen our understanding of human cognitive processes.
Diksha Bhandari (Thu,) studied this question.