Thesis Defenses Seminar
Thursday, July 6, 2023, 9:00am, Zoom
Tao Tang (Duke University, Mathematics)
Three essays of Bayesian inferences on dynamical systems, continuous time Markov chains, and low-dimensional structure
Abstract:
This dissertation propels the frontier of Bayesian inference, particularly for structurally diverse data, by developing innovative methodologies and robust theories, delineated into three distinct segments, each tackling Dynamical systems, continuous time Markov Chains, and data with low-dimensional structures. The initial part presents the Hierarchical Shrinkage Gaussian Process (HierGP), a novel model adept at detecting structured sparse features within limited data from response surfaces. Incorporating the principles of effect sparsity, heredity, and hierarchy into a Gaussian process framework, HierGP demonstrates superior performance compared to its contemporaries. This superiority is backed by a range of numerical experiments and its successful application to dynamical system recovery. The dissertation's second segment introduces the Bayesian Approximation Spectral Inference (BASI) method. Inspired by the necessity for pragmatic inference techniques for continuous-time Markov chains, BASI innovatively employs probabilistic matrix factorization to estimate a continuous-time Markov chain generator's low-rank representation. This approach effectively circumvents the computational challenges of assessing implicit likelihood functions inherent in previous methods, with our theoretical support for BASI showcased via the asymptotic properties of the posterior distribution. In its final part, the dissertation delves into posterior contraction rates for Gaussian process regression in the context of data with low-dimensional structures. The conditions established enable adaptivity to any intrinsic dimension, thereby giving the optimal posterior contraction rate. This exploration is further enriched by an innovative empirical Bayes prior of bandwidth, dispensing with the necessity for prior knowledge of the intrinsic dimension. Cumulatively, these three essays enhance Bayesian statistics for data with varying and intricate structures. The dissertation seamlessly intertwines theoretical contribution with practical numerical experiments, lending empirical weight to our innovative propositions.

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