Optimal Transport, Convexity and Mean Field Variational Bayesian Approximation

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Sadeghi, Saeed | 2024

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Type of Document: Ph.D. Dissertation
Language: Farsi
Document No: 57429 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Bahraini, Alireza
Abstract:
We intend to find conditions that lead to a posterior probability distribution becoming convex in terms of $-\log$. To achieve this, we will leverage modern optimal transport tools and their geometric results. Then, in relevant examples related to the theory of Bayesian mean field approximation, we aim to enhance convex models, enabling more precise and improved results. We do this task in the case of Gaussian mixture models. Also, we introduce a new definition of convexity on discrete spaces and strive to utilize this definition for the aforementioned estimation. Additionally, the presented definition can be studied independently
Keywords:
Continuity Equation ; Gaussian Mixture Model ; Optimal Transport ; Ricci Curvature ; Mean-Field Theory ; Probability Measure Space ; Geodesic Convexity ; Discrete Convexity