# 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 “What 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 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. arXiv:2005.12998 [Math], January.
Alexanderian, Alen, Noemi Petra, Georg Stadler, and Omar Ghattas. 2016. SIAM Journal on Scientific Computing 38 (1): A243–72.
Borgonovo, E., W. Castaings, and S. Tarantola. 2012. 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. Journal of the American Statistical Association 88 (421): 9–25.
Bui-Thanh, Tan. 2012.
Dashti, Masoumeh, and Andrew M. Stuart. 2015. arXiv:1302.6989 [Math], July.
Foong, Andrew Y. K., Yingzhen Li, José Miguel Hernández-Lobato, and Richard E. Turner. 2019. arXiv:1906.11537 [Cs, Stat], June.
Giordano, Matteo, and Richard Nickl. 2020. Inverse Problems 36 (8): 085001.
Immer, Alexander, Maciej Korzepa, and Matthias Bauer. 2021. 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. Journal of Computational and Applied Mathematics 198 (2): 493–504.
Kennedy, Marc C., and Anthony O’Hagan. 2001. 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. The Annals of Statistics 39 (5).
Mosegaard, Klaus, and Albert Tarantola. 1995. Journal of Geophysical Research: Solid Earth 100 (B7): 12431–47.
———. 2002. In International Geophysics, 81:237–65. Elsevier.
O’Hagan, A. 2006. Reliability Engineering & System Safety, The Fourth International Conference on Sensitivity Analysis of Model Output (SAMO 2004), 91 (10): 1290–300.
Plumlee, Matthew. 2017. Journal of the American Statistical Association 112 (519): 1274–85.
Särkkä, Simo, A. Solin, and J. Hartikainen. 2013. IEEE Signal Processing Magazine 30 (4): 51–61.
Schillings, Claudia, and Andrew M. Stuart. 2017. SIAM Journal on Numerical Analysis 55 (3): 1264–90.
Schwab, C., and A. M. Stuart. 2012. Inverse Problems 28 (4): 045003.
Stuart, A. M. 2010. Acta Numerica 19: 451–559.
Tait, Daniel J., and Theodoros Damoulas. 2020. arXiv:2006.15641 [Cs, Stat], June.
Tarantola, Albert. 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM.
———. n.d. Mapping Of Probabilities.
Tonolini, Francesco, Jack Radford, Alex Turpin, Daniele Faccio, and Roderick Murray-Smith. 2020. Journal of Machine Learning Research 21 (179): 1–46.
Wacker, Philipp. 2017. arXiv:1701.07989 [Math], April.
Wei, Qi, Kai Fan, Lawrence Carin, and Katherine A. Heller. 2017. arXiv:1709.01841 [Cs], September.
Yang, Liu, Xuhui Meng, and George Em Karniadakis. 2021. 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. 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.↩︎

### No comments yet. Why not leave one?

GitHub-flavored Markdown & a sane subset of HTML is supported.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8937525749206543, "perplexity": 3417.5232088646944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710972.37/warc/CC-MAIN-20221204104311-20221204134311-00033.warc.gz"}