Publications

Dual Likelihood for Causal Inference under Structure Uncertainty

Published in Proceedings of Machine Learning Research, 2024

Abstract: Knowledge of the underlying causal relations is essential for inferring the effect of interventions in complex systems. In a widely studied approach, structural causal models postulate noisy functional relations among interacting variables, where the underlying causal structure is then naturally represented by a directed graph whose edges indicate direct causal dependencies. In the typical application, this underlying causal structure must be learned from data, and thus, the remaining structure uncertainty needs to be incorporated into causal inference in order to draw reliable conclusions. In recent work, test inversions provide an ansatz to account for this data-driven model choice and, therefore, combine structure learning with causal inference. In this article, we propose the use of dual likelihood to greatly simplify the treatment of the involved testing problem. Indeed, dual likelihood leads to a closed-form solution for constructing confidence regions for total causal effects that rigorously capture both sources of uncertainty: causal structure and numerical size of nonzero effects. The proposed confidence regions can be computed with a bottom-up procedure starting from sink nodes. To render the causal structure identifiable, we develop our ideas in the context of linear causal relations with equal error variances.

Recommended citation: D. Strieder and M. Drton. Dual Likelihood for Causal Inference under Structure Uncertainty. Proceedings of the Third Conference on Causal Learning and Reasoning, PMLR 236:1-17, (2024). https://proceedings.mlr.press/v236/strieder24a.html

Confidence in causal inference under structure uncertainty in linear causal models with equal variances

Published in Journal of Causal Inference, 2023

Abstract: Inferring the effect of interventions within complex systems is a fundamental problem of statistics. A widely studied approach uses structural causal models that postulate noisy functional relations among a set of interacting variables. The underlying causal structure is then naturally represented by a directed graph whose edges indicate direct causal dependencies. In a recent line of work, additional assumptions on the causal models have been shown to render this causal graph identifiable from observational data alone. One example is the assumption of linear causal relations with equal error variances that we will take up in this work. When the graph structure is known, classical methods may be used for calculating estimates and confidence intervals for causal-effects. However, in many applications, expert knowledge that provides an a priori valid causal structure is not available. Lacking alternatives, a commonly used two-step approach first learns a graph and then treats the graph as known in inference. This, however, yields confidence intervals that are overly optimistic and fail to account for the data-driven model choice. We argue that to draw reliable conclusions, it is necessary to incorporate the remaining uncertainty about the underlying causal structure in confidence statements about causal-effects. To address this issue, we present a framework based on test inversion that allows us to give confidence regions for total causal-effects that capture both sources of uncertainty: causal structure and numerical size of non-zero effects.

Recommended citation: D. Strieder and M. Drton. Confidence in causal inference under structure uncertainty in linear causal models with equal variances. Journal of Causal Inference, 11(1), 0030, (2023). https://doi.org/10.1515/jci-2023-0030

Rank-based causal discovery for post-nonlinear models

Published in Proceedings of Machine Learning Research, 2023

Abstract: Learning causal relationships from empirical observations is a central task in scientific research. A common method is to employ structural causal models that postulate noisy functional relations among a set of interacting variables. To ensure unique identifiability of causal directions, researchers consider restricted subclasses of structural causal models. Post-nonlinear (PNL) causal models constitute one of the most flexible options for such restricted subclasses, containing in particular the popular additive noise models as a further subclass. However, learning PNL models is not well studied beyond the bivariate case. The existing methods learn non-linear functional relations by minimizing residual dependencies and subsequently test independence from residuals to determine causal orientations. However, these methods can be prone to overfitting and, thus, difficult to tune appropriately in practice. As an alternative, we propose a new approach for PNL causal discovery that uses rank-based methods to estimate the functional parameters. This new approach exploits natural invariances of PNL models and disentangles the estimation of the non-linear functions from the independence tests used to find causal orientations. We prove consistency of our method and validate our results in numerical experiments.

Recommended citation: G. Keropyan, D. Strieder and M. Drton. Rank-Based Causal Discovery for Post-Nonlinear Models Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, PMLR 206:7849-7870, (2023). https://proceedings.mlr.press/v206/keropyan23a.html

On the choice of the splitting ratio for the split likelihood ratio test

Published in Electronic Journal of Statistics, 2022

Abstract: The recently introduced framework of universal inference provides a new approach to constructing hypothesis tests and confidence regions that are valid in finite samples and do not rely on any specific regularity assumptions on the underlying statistical model. At the core of the methodology is a split likelihood ratio statistic, which is formed under data splitting and compared to a cleverly selected universal critical value. As this critical value can be very conservative, it is interesting to mitigate the potential loss of power by careful choice of the ratio according to which data are split. Motivated by this problem, we study the split likelihood ratio test under local alternatives and introduce the resulting class of noncentral split chi-square distributions. We investigate the properties of this new class of distributions and use it to numerically examine and propose an optimal choice of the data splitting ratio for tests of composite hypotheses of different dimensions.

Recommended citation: D. Strieder and M. Drton. On the choice of the splitting ratio for the split likelihood ratio test. Electronic Journal of Statistics, 16(2), 6631-6650, (2022). https://doi.org/10.1214/22-EJS2099

Confidence in causal discovery with linear causal models

Published in Proceedings of Machine Learning Research, 2021

Abstract: Structural causal models postulate noisy functional relations among a set of interacting variables. The causal structure underlying each such model is naturally represented by a directed graph whose edges indicate for each variable which other variables it causally depends upon. Under a number of different model assumptions, it has been shown that this causal graph and, thus also, causal effects are identifiable from mere observational data. For these models, practical algorithms have been devised to learn the graph. Moreover, when the graph is known, standard techniques may be used to give estimates and confidence intervals for causal effects. We argue, however, that a two-step method that first learns a graph and then treats the graph as known yields confidence intervals that are overly optimistic and can drastically fail to account for the uncertain causal structure. To address this issue we lay out a framework based on test inversion that allows us to give confidence regions for total causal effects that capture both sources of uncertainty: causal structure and numerical size of nonzero effects. Our ideas are developed in the context of bivariate linear causal models with homoscedastic errors, but as we exemplify they are generalizable to larger systems as well as other settings such as, in particular, linear non-Gaussian models.

Recommended citation: D. Strieder, T. Freidling, S. Haffner and M. Drton. Confidence in Causal Discovery with Linear Causal Models. Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence, PMLR 161:1217-1226, (2021). https://proceedings.mlr.press/v161/strieder21a.html

Testing normality in any dimension by Fourier methods in a multivariate Stein equation

Published in Canadian Journal of Statistics, 2021

Abstract: We study a novel class of affine-invariant and consistent tests for multivariate normality. The tests are based on a characterization of the standard d-variate normal distribution by way of the unique solution of an initial value problem connected to a partial differential equation, which is motivated by a multivariate Stein equation. The test criterion is a suitably weighted L2-statistic. We derive the limit distribution of the test statistic under the null hypothesis as well as under contiguous and fixed alternatives to normality. A consistent estimator of the limiting variance under fixed alternatives, as well as an asymptotic confidence interval of the distance of an underlying alternative with respect to the multivariate normal law, is derived. In simulation studies, we show that the tests are strong in comparison with prominent competitors and that the empirical coverage rate of the asymptotic confidence interval converges to the nominal level. We present a real data example and also outline topics for further research.

Recommended citation: B. Ebner, N. Henze and D.Strieder. Testing normality in any dimension by Fourier methods in a multivariate Stein equation. Canadian Journal of Statistics, 50: 992-1033, (2022). https://doi.org/10.1002/cjs.11670

David Strieder