The following main results have been achieved which are classified into different sub-areas.
(1) Causal regularization and corresponding distributional robustness. We achieved a major result about the duality of causal regularization, encouraging certain invariances over different sub-populations, and distributional robustness with respect to worst case expected loss when the test data distribution is perturbed: this is developed in Rothenhäusler et al. (2021) for linear (structural equation model) systems and extended and put into broader context with nonlinear models in a discussion paper in Statistical Science (Bühlmann, 2020a,b). Also, the invariance idea has been successfully transferred to the area of blind deconvolution with independence component analysis: the paper by Pfister et al. (2019) contains the novel methodology and theory and demonstrates its use for brain EEG signal processing and a deconvolution problem in climate science. The work around invariance and causality has also appeared in a perspective article in PNAS (Bühlmann, 2020c).
(2) Deconfounding in high-dimensional problems. We achieved a strong result how to adjust for unobserved latent and dense confounding in high-dimensional linear models and nonlinear models. The technique is based on spectral transformations of the data, by transforming the singular values of the matrix of covariates. The method is shown to achieve the same convergence rate for statistical accuracy as if there were no hidden confounders. The implications and potential applications are wide-ranging, essentially for any high-dimensional data analysis (e.g. in genetics, genomics or climate science). The first main methodological and mathematical contribution is published in Cevid et al. (2020). In Guo et al. (2022), we advance the method for uncertainty quantification: again, under a dense confoudnig assumption one can do asymptotically as well as without confounding variables. The main contribution has been brought into broader context by Bühlmann and Cevid (2020).
(3) Massive computational gains in change point detection. At the heart of many problems with the analysis of heterogeneous data is the segmentation or partitioning into homogeneous parts. We have developed massive computational speed-ups for such partitioning, known as change point detection. The different methods including software are worked out in Londschien et al. (2020, 2023), and Kovacs et al. (2020, 2023).
(4) Statistical machine learning. We made contributions to double machine learning (Emmenegger and Bühlmann, 2021, 2023) and we created the Distributional Random Forests algorithm (Cevid et al., 2022) and analyzed its further mathematical properties for statistical inference (Näf et al., 2023).
(5) Goodness of fit of causal models. We pioneered new approaches for goodness of fit for linear and nonlinear causal models (Schultheiss et al. 2023; Schultheiss and Bühlmann 2023a, 2023b, 2024).
(6) Related interdisciplinary work. Causal invariance techniques have been used in Pfister et al. (2021) and Williams et al. (2022) for the molecular landscape of the aging mouse; deconfounding has been used in Jablonski et al. (2022) for identifying cancer pathway dysregulation.
A breakthrough result has been achieved by Yuansi Chen, by then a postdoc paid by the ERC grant, who "essentially" solved the KLS conjecture (Chen 2021), see also
https://www.quantamagazine.org/statistics-postdoc-tames-decades-old-geometry-problem-20210301/(se abrirá en una nueva ventana) This result came as a surprise and has nothing to do with the formulated goals of the project. However, this demonstrates impressively that outstanding statisticians and mathematicians are able to contribute beyondt their area of original expertise!
