Bayesian inverse problems



Inverse problem solution with a probabilistic approach. In Bayesian terms, say we have a model which gives us the density of a certain output observation \(y\) for a given input \(x\) which we write as \(p(y\mid x)\). By Bayes’ rule we can find the density of inputs for a given observed output by \[p(x \mid y)=\frac{p(x) p(y \mid x)}{p(y)}.\] The process of computing \[p(x \mid y)\] is the most basic step of Bayesian inference, nothing special to see here.

In the world I live in, \(p(y \mid x)\) is not completely specified, but is a regression density with unknown parameters \(\theta\) that we must also learn, that may have prior densities of their own. Maybe I also wish to parameterise the density for the prior on \(x\), \(p(x \mid \lambda),\) which is typically independent of \(\theta.\) Now the model is a hierarchical Bayes model, leading to a directed factorisation \[p(x,y,\theta,\lambda)=p(\theta)p(\lambda)p(x\mid \lambda) p(y\mid x,\theta).\] We can use more Bayes rule to write the density of interest as \[p(x, \theta, \lambda \mid y) \propto p(y \mid x, \theta)p(x \mid\lambda)p(\lambda)p(\theta).\] Solving this is also, I believe, sometimes called joint inversion. For my applications, we usually want to do this in two phases. In the first, we have some data set of \(N\) input-output pairs indexed by \(i,\) \(\mathcal{D}=\{(x_i, y_i:i=1,\dots,N)\}\) which we use to estimate posterior density \(p(\theta,\lambda \mid \mathcal{D})\) in some learning phase. Thereafter we only ever wish to find \(p(x, \theta, \lambda \mid y,\mathcal{D})\) or possibly even \(p(x \mid y,\mathcal{D})\) but either way do not thereafter update \(\theta, \lambda|\mathcal{D}\).

If the problem is high dimensional, in the sense that \(x\in \mathbb{R}^n\) for \(n\) large and ill-posed, in the sense that, e.g. \(y\in\mathbb{R}^m\) with \(n>m\), we have a particular set of challenges which it is useful to group under the heading of functional inverse problems.1 A classic example of this class of problem is β€œWhat was the true image what was blurred to create this corrupted version?”.

Laplace method

We can use Laplace approximation approximate latent density.

Laplace approximations seems like it might have an attractive feature: providing estimates also for inverse problems (Breslow and Clayton 1993; Wacker 2017; Alexanderian et al. 2016; Alexanderian 2021) by leveraging the delta method. I think this should come out nice in network linearization approaches such as Foong et al. (2019) and Immer, Korzepa, and Bauer (2021).

Suppose we have a regression network that outputs (perhaps approximately) a Gaussian distribution for outputs given inputs.

TBC.

References

Alexanderian, Alen. 2021. β€œOptimal Experimental Design for Infinite-Dimensional Bayesian Inverse Problems Governed by PDEs: A Review.” arXiv:2005.12998 [Math], January.
Alexanderian, Alen, Noemi Petra, Georg Stadler, and Omar Ghattas. 2016. β€œA Fast and Scalable Method for A-Optimal Design of Experiments for Infinite-Dimensional Bayesian Nonlinear Inverse Problems.” SIAM Journal on Scientific Computing 38 (1): A243–72.
Borgonovo, E., W. Castaings, and S. Tarantola. 2012. β€œModel Emulation and Moment-Independent Sensitivity Analysis: An Application toΒ Environmental Modelling.” Environmental Modelling & Software, Emulation techniques for the reduction and sensitivity analysis of complex environmental models, 34 (June): 105–15.
Breslow, N. E., and D. G. Clayton. 1993. β€œApproximate Inference in Generalized Linear Mixed Models.” Journal of the American Statistical Association 88 (421): 9–25.
Bui-Thanh, Tan. 2012. β€œA Gentle Tutorial on Statistical Inversion Using the Bayesian Paradigm.”
Dashti, Masoumeh, and Andrew M. Stuart. 2015. β€œThe Bayesian Approach To Inverse Problems.” arXiv:1302.6989 [Math], July.
Foong, Andrew Y. K., Yingzhen Li, JosΓ© Miguel HernΓ‘ndez-Lobato, and Richard E. Turner. 2019. β€œβ€˜In-Between’ Uncertainty in Bayesian Neural Networks.” arXiv:1906.11537 [Cs, Stat], June.
Giordano, Matteo, and Richard Nickl. 2020. β€œConsistency of Bayesian Inference with Gaussian Process Priors in an Elliptic Inverse Problem.” Inverse Problems 36 (8): 085001.
Immer, Alexander, Maciej Korzepa, and Matthias Bauer. 2021. β€œImproving Predictions of Bayesian Neural Nets via Local Linearization.” In International Conference on Artificial Intelligence and Statistics, 703–11. PMLR.
Kaipio, Jari, and E. Somersalo. 2005. Statistical and Computational Inverse Problems. Applied Mathematical Sciences. New York: Springer-Verlag.
Kaipio, Jari, and Erkki Somersalo. 2007. β€œStatistical Inverse Problems: Discretization, Model Reduction and Inverse Crimes.” Journal of Computational and Applied Mathematics 198 (2): 493–504.
Kennedy, Marc C., and Anthony O’Hagan. 2001. β€œBayesian Calibration of Computer Models.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 63 (3): 425–64.
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).
Mosegaard, Klaus, and Albert Tarantola. 1995. β€œMonte Carlo Sampling of Solutions to Inverse Problems.” Journal of Geophysical Research: Solid Earth 100 (B7): 12431–47.
β€”β€”β€”. 2002. β€œProbabilistic Approach to Inverse Problems.” In International Geophysics, 81:237–65. Elsevier.
O’Hagan, A. 2006. β€œBayesian Analysis of Computer Code Outputs: A Tutorial.” Reliability Engineering & System Safety, The Fourth International Conference on Sensitivity Analysis of Model Output (SAMO 2004), 91 (10): 1290–300.
Plumlee, Matthew. 2017. β€œBayesian Calibration of Inexact Computer Models.” Journal of the American Statistical Association 112 (519): 1274–85.
SΓ€rkkΓ€, Simo, A. Solin, and J. Hartikainen. 2013. β€œSpatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.” IEEE Signal Processing Magazine 30 (4): 51–61.
Schillings, Claudia, and Andrew M. Stuart. 2017. β€œAnalysis of the Ensemble Kalman Filter for Inverse Problems.” SIAM Journal on Numerical Analysis 55 (3): 1264–90.
Schwab, C., and A. M. Stuart. 2012. β€œSparse Deterministic Approximation of Bayesian Inverse Problems.” Inverse Problems 28 (4): 045003.
Stuart, A. M. 2010. β€œInverse Problems: A Bayesian Perspective.” Acta Numerica 19: 451–559.
Tait, Daniel J., and Theodoros Damoulas. 2020. β€œVariational Autoencoding of PDE Inverse Problems.” arXiv:2006.15641 [Cs, Stat], June.
Tarantola, Albert. 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM.
β€”β€”β€”. n.d. Mapping Of Probabilities.
Tonolini, Francesco, Jack Radford, Alex Turpin, Daniele Faccio, and Roderick Murray-Smith. 2020. β€œVariational Inference for Computational Imaging Inverse Problems.” Journal of Machine Learning Research 21 (179): 1–46.
Wacker, Philipp. 2017. β€œLaplace’s Method in Bayesian Inverse Problems.” arXiv:1701.07989 [Math], April.
Wei, Qi, Kai Fan, Lawrence Carin, and Katherine A. Heller. 2017. β€œAn Inner-Loop Free Solution to Inverse Problems Using Deep Neural Networks.” arXiv:1709.01841 [Cs], September.
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.
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.

  1. There is also a strand of the literature which refers to any form of Bayesian inference as an inverse problem, but this usage does not draw a helpful distinction for me so I avoid it.β†©οΈŽ


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