Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems

• Applicant: Professor Timothy Sullivan, Ph.D.
• Subject Area: Mathematics
• Term: from 2019 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 415980428

• Final Report Year: 2023

Final Report Abstract

This project aimed to advance the state of the art in the mathematical understanding of Bayesian inverse problems (BIPs), a common framework for statistical learning, by establishing convergence and stability results for maximum a posteriori (MAP) estimators. A single postdoctoral researcher was funded for two years. The research team was able to establish novel stability results using the framework of Γ-convergence of Onsager–Machlup functionals for the Bayesian posterior measures and these results were accepted for publication in the leading international journal for the field of inverse problems. Further lines of research that are still under investigation.

Publications

• Γ -convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems. Inverse Problems, 38(2), 025005.
  Ayanbayev; Birzhan; Klebanov; Ilja; Lie; Han, Cheng; Sullivan & T. J.
  
• Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces. Inverse Problems, 38(2), 025006.
  Ayanbayev, Birzhan; Klebanov, Ilja; Lie, Han Cheng & Sullivan, T. J.
  
• An Order-Theoretic Perspective on Modes and Maximum A Posteriori Estimation in Bayesian Inverse Problems. SIAM/ASA Journal on Uncertainty Quantification, 11(4), 1195-1224.
  Lambley, Hefin & Sullivan, T. J.