Bayes functional regression



Junction for various bayesian methods where the estimands are functions over some sintunuous argument space.

Gaussian process regression

See Gaussian process regression.

On manifolds

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

πŸ—

Neural processes

See neural processes.

Generic nonparametrics

See Bayes nonparametrics.

References

Alexanderian, Alen. 2021. β€œOptimal 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. β€œSparse 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. β€œA 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. β€œBesov Priors for Bayesian Inverse Problems.” arXiv.
Dashti, Masoumeh, and Andrew M. Stuart. 2015. β€œThe Bayesian Approach To Inverse Problems.” arXiv:1302.6989 [Math], July.
Dubrule, Olivier. 2018. β€œKriging, 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 SΓ€rkkΓ€. 2017. β€œParallelizable Sparse Inverse Formulation Gaussian Processes (SpInGP).” In arXiv:1610.08035 [Stat].
Jo, Hyeontae, Hwijae Son, Hyung Ju Hwang, and Eun Heui Kim. 2020. β€œDeep 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. β€œBayesian Inverse Problems with Gaussian Priors.” The Annals of Statistics 39 (5).
Lasanen, S, and L Roininen. 2005. β€œStatistical 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. β€œCan One Use Total Variation Prior for Edge-Preserving Bayesian Inversion?” Inverse Problems 20 (5): 1537–63.
Lee, Kuang-Yao, Dingjue Ji, Lexin Li, Todd Constable, and Hongyu Zhao. 2023. β€œConditional Functional Graphical Models.” Journal of the American Statistical Association 118 (541): 257–71.
Liu, Xiao, Kyongmin Yeo, and Siyuan Lu. 2020. β€œStatistical 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. β€œThe Functional Neural Process.” In Advances in Neural Information Processing Systems. Vol. 32. Curran Associates, Inc.
Magnani, Emilia, Nicholas KrΓ€mer, Runa Eschenhagen, Lorenzo Rosasco, and Philipp Hennig. 2022. β€œApproximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs.” arXiv.
Mosegaard, Klaus, and Albert Tarantola. 2002. β€œProbabilistic Approach to Inverse Problems.” In International Geophysics, 81:237–65. Elsevier.
Perdikaris, Paris, and George Em Karniadakis. 2016. β€œModel 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. β€œA 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. β€œSpectral Diffusion Processes.” In.
Pielok, Tobias, Bernd Bischl, and David RΓΌgamer. 2023. β€œApproximate Bayesian Inference with Stein Functional Variational Gradient Descent.” In.
Pikkarainen, Hanna Katriina. 2006. β€œState 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. β€œKullback-Leibler Approximation for Probability Measures on Infinite Dimensional Spaces.” SIAM Journal on Mathematical Analysis 47 (6): 4091–4122.
Sigrist, Fabio, Hans R. KΓΌnsch, and Werner A. Stahel. 2015. β€œSpate : 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. β€œSequential Neural Processes.” arXiv:1906.10264 [Cs, Stat], June.
Song, Yang, Liyue Shen, Lei Xing, and Stefano Ermon. 2022. β€œSolving 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. β€œScore-Based Generative Modeling Through Stochastic Differential Equations.” In.
Sun, Shengyang, Guodong Zhang, Jiaxin Shi, and Roger Grosse. 2019. β€œFunctional Variational Bayesian Neural Networks.” In.
Tran, Ba-Hien, Simone Rossi, Dimitrios Milios, and Maurizio Filippone. 2022. β€œAll You Need Is a Good Functional Prior for Bayesian Deep Learning.” Journal of Machine Learning Research 23 (74): 1–56.
Unser, M. 2015. β€œSampling 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. β€œA 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. β€œA Unified Formulation of Gaussian Vs Sparse Stochastic Processesβ€”Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory 60 (3): 1945–62.
Valentine, Andrew P, and Malcolm Sambridge. 2020a. β€œGaussian Process Modelsβ€”I. A Framework for Probabilistic Continuous Inverse Theory.” Geophysical Journal International 220 (3): 1632–47.
β€”β€”β€”. 2020b. β€œGaussian Process Modelsβ€”II. Lessons for Discrete Inversion.” Geophysical Journal International 220 (3): 1648–56.
Wang, Ziyu, Tongzheng Ren, Jun Zhu, and Bo Zhang. 2018. β€œFunction Space Particle Optimization for Bayesian Neural Networks.” In.
Watson, Joe, Jihao Andreas Lin, Pascal Klink, and Jan Peters. 2020. β€œNeural Linear Models with Functional Gaussian Process Priors.” In, 10.
Yang, Liu, Xuhui Meng, and George Em Karniadakis. 2021. β€œB-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. β€œPhysics-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. β€œWOMBAT 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. β€œLearning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks.” SIAM Journal on Scientific Computing 42 (2): A639–65.

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