Junction for various bayesian methods where the estimands are functions over some sintunuous argument space.
Gaussian process regression
I would like to read Terenin on GPs on Manifolds who also makes a suggestive connection to SDEs, which is the filtering GPs trick again.
By variational inference
See neural processes.
See Bayes nonparametrics.
Alexanderian, Alen. 2021. “Optimal Experimental Design for Infinite-Dimensional Bayesian Inverse Problems Governed by PDEs: A Review.” arXiv:2005.12998 [Math], January.
Bostan, E., U. S. Kamilov, M. Nilchian, and M. Unser. 2013. “Sparse Stochastic Processes and Discretization of Linear Inverse Problems.” IEEE Transactions on Image Processing 22 (7): 2699–2710.
Bui-Thanh, Tan, Omar Ghattas, James Martin, and Georg Stadler. 2013. “A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion.” SIAM Journal on Scientific Computing 35 (6): A2494–2523.
Dashti, Masoumeh, Stephen Harris, and Andrew Stuart. 2011. “Besov Priors for Bayesian Inverse Problems.” arXiv.
Dashti, Masoumeh, and Andrew M. Stuart. 2015. “The Bayesian Approach To Inverse Problems.” arXiv:1302.6989 [Math], July.
Dubrule, Olivier. 2018. “Kriging, Splines, Conditional Simulation, Bayesian Inversion and Ensemble Kalman Filtering.” In Handbook of Mathematical Geosciences: Fifty Years of IAMG, edited by B.S. Daya Sagar, Qiuming Cheng, and Frits Agterberg, 3–24. Cham: Springer International Publishing.
Grigorievskiy, Alexander, Neil Lawrence, and Simo Särkkä. 2017. “Parallelizable Sparse Inverse Formulation Gaussian Processes (SpInGP).” In arXiv:1610.08035 [Stat].
Jo, Hyeontae, Hwijae Son, Hyung Ju Hwang, and Eun Heui Kim. 2020. “Deep Neural Network Approach to Forward-Inverse Problems.” Networks & Heterogeneous Media 15 (2): 247.
Knapik, B. T., A. W. van der Vaart, and J. H. van Zanten. 2011. “Bayesian Inverse Problems with Gaussian Priors.” The Annals of Statistics 39 (5).
Lasanen, S, and L Roininen. 2005. “Statistical Inversion with Green’s Priors.” In Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11.
Lassas, Matti, and Samuli Siltanen. 2004. “Can One Use Total Variation Prior for Edge-Preserving Bayesian Inversion?” Inverse Problems 20 (5): 1537–63.
Liu, Xiao, Kyongmin Yeo, and Siyuan Lu. 2020. “Statistical Modeling for Spatio-Temporal Data From Stochastic Convection-Diffusion Processes.” Journal of the American Statistical Association 0 (0): 1–18.
Louizos, Christos, Xiahan Shi, Klamer Schutte, and Max Welling. 2019. “The Functional Neural Process.” In Advances in Neural Information Processing Systems. Vol. 32. Curran Associates, Inc.
Magnani, Emilia, Nicholas Krämer, Runa Eschenhagen, Lorenzo Rosasco, and Philipp Hennig. 2022. “Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs.” arXiv.
Mosegaard, Klaus, and Albert Tarantola. 2002. “Probabilistic Approach to Inverse Problems.” In International Geophysics, 81:237–65. Elsevier.
Perdikaris, Paris, and George Em Karniadakis. 2016. “Model inversion via multi-fidelity Bayesian optimization: a new paradigm for parameter estimation in haemodynamics, and beyond.” Journal of the Royal Society, Interface 13 (118): 20151107.
Petra, Noemi, James Martin, Georg Stadler, and Omar Ghattas. 2014. “A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems, Part II: Stochastic Newton MCMC with Application to Ice Sheet Flow Inverse Problems.” SIAM Journal on Scientific Computing 36 (4): A1525–55.
Phillips, Angus, Thomas Seror, Michael John Hutchinson, Valentin De Bortoli, Arnaud Doucet, and Emile Mathieu. 2022. “Spectral Diffusion Processes.” In.
Pielok, Tobias, Bernd Bischl, and David Rügamer. 2023. “Approximate Bayesian Inference with Stein Functional Variational Gradient Descent.” In.
Pikkarainen, Hanna Katriina. 2006. “State Estimation Approach to Nonstationary Inverse Problems: Discretization Error and Filtering Problem.” Inverse Problems 22 (1): 365–79.
Pinski, F. J., G. Simpson, A. M. Stuart, and H. Weber. 2015. “Kullback-Leibler Approximation for Probability Measures on Infinite Dimensional Spaces.” SIAM Journal on Mathematical Analysis 47 (6): 4091–4122.
Sigrist, Fabio, Hans R. Künsch, and Werner A. Stahel. 2015. “Spate : An R Package for Spatio-Temporal Modeling with a Stochastic Advection-Diffusion Process.” Application/pdf. Journal of Statistical Software 63 (14).
Singh, Gautam, Jaesik Yoon, Youngsung Son, and Sungjin Ahn. 2019. “Sequential Neural Processes.” arXiv:1906.10264 [Cs, Stat], June.
Song, Yang, Liyue Shen, Lei Xing, and Stefano Ermon. 2022. “Solving Inverse Problems in Medical Imaging with Score-Based Generative Models.” In. arXiv.
Song, Yang, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. 2022. “Score-Based Generative Modeling Through Stochastic Differential Equations.” In.
Sun, Shengyang, Guodong Zhang, Jiaxin Shi, and Roger Grosse. 2019. “Functional Variational Bayesian Neural Networks.” In.
Tran, Ba-Hien, Simone Rossi, Dimitrios Milios, and Maurizio Filippone. 2022. “All You Need Is a Good Functional Prior for Bayesian Deep Learning.” Journal of Machine Learning Research 23 (74): 1–56.
Unser, M. 2015. “Sampling and (Sparse) Stochastic Processes: A Tale of Splines and Innovation.” In 2015 International Conference on Sampling Theory and Applications (SampTA), 221–25.
Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press.
Unser, M., P. D. Tafti, A. Amini, and H. Kirshner. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” IEEE Transactions on Information Theory 60 (5): 3036–51.
Unser, M., P. D. Tafti, and Q. Sun. 2014. “A Unified Formulation of Gaussian Vs Sparse Stochastic Processes—Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory 60 (3): 1945–62.
Valentine, Andrew P, and Malcolm Sambridge. 2020a. “Gaussian Process Models—I. A Framework for Probabilistic Continuous Inverse Theory.” Geophysical Journal International 220 (3): 1632–47.
———. 2020b. “Gaussian Process Models—II. Lessons for Discrete Inversion.” Geophysical Journal International 220 (3): 1648–56.
Wang, Ziyu, Tongzheng Ren, Jun Zhu, and Bo Zhang. 2018. “Function Space Particle Optimization for Bayesian Neural Networks.” In.
Watson, Joe, Jihao Andreas Lin, Pascal Klink, and Jan Peters. 2020. “Neural Linear Models with Functional Gaussian Process Priors.” In, 10.
Yang, Liu, Xuhui Meng, and George Em Karniadakis. 2021. “B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data.” Journal of Computational Physics 425 (January): 109913.
Yang, Liu, Dongkun Zhang, and George Em Karniadakis. 2020. “Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations.” SIAM Journal on Scientific Computing 42 (1): A292–317.
Zammit-Mangion, Andrew, Michael Bertolacci, Jenny Fisher, Ann Stavert, Matthew L. Rigby, Yi Cao, and Noel Cressie. 2021. “WOMBAT v1.0: A fully Bayesian global flux-inversion framework.” Geoscientific Model Development Discussions, July, 1–51.
Zhang, Dongkun, Ling Guo, and George Em Karniadakis. 2020. “Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks.” SIAM Journal on Scientific Computing 42 (2): A639–65.
No comments yet. Why not leave one?