arXiv subjects
Publications and source records attributed to Peter Bühlmann.
We review some recent work on removing hidden confounding and causal regularization from a unified viewpoint. We describe how simple and user-friendly techniques improve stability, replicability and distributional robustness in heterogeneous data. In this sense, we provide additional thoughts to the issue on concept drift, raised by Efron (2020), when the data generating distribution is changing.
In this discussion, we compare the choice of seeded intervals and that of random intervals for change point segmentation from practical, statistical and computational perspectives. Furthermore, we investigate a novel estimator of the noise level, which improves many existing model selection procedures (including the steepest drop to low levels), particularly for challenging frequent change point scenarios with low signal-to-noise ratios.
We consider the problem of predicting a response variable from a set of covariates on a data set that differs in distribution from the training data. Causal parameters are optimal in terms of predictive accuracy if in the new distribution either many variables are affected by interventions or only some variables are affected, but the perturbations are strong. If the training and test distributions differ by a shift, causal parameters might be too conservative to perform well on the above task. This motivates anchor regression, a method that makes use of exogeneous variables to solve a relaxation of the causal minimax problem by considering a modification of the least-squares loss. The procedure naturally provides an interpolation between the solutions of ordinary least squares and two-stage least squares. We prove that the estimator satisfies predictive guarantees in terms of distributional robustness against shifts in a linear class; these guarantees are valid even if the instrumental variables assumptions are violated. If anchor regression and least squares provide the same answer (anchor stability), we establish that OLS parameters are invariant under certain distributional changes. Anchor regression is shown empirically to improve replicability and protect against distributional shifts.
Randomized trials and observational studies, more often than not, run over a certain period of time. The treatment effect evolves during this period which provides crucial insights into the treatment response and the long-term effects. Many conventional methods for estimating treatment effects are limited to the i.i.d. setting and are not suited for inferring the time dynamics of the treatment effect. The time series encountered in these settings are highly informative but often nonstationary due to the changing effects of treatment. This increases the difficulty, since stationarity, a common assumption in time series analysis, cannot be reasonably assumed. Another challenge is the heterogeneity of the treatment effect when the treatment affects units differently. The task of estimating heterogeneous treatment effects from nonstationary and, in particular, interventional time series is highly relevant but has remained unexplored yet. We propose Causal Transfer, a method which combines regression to adjust for confounding with time series modelling to learn the effect of the treatment and how it evolves over time. Causal Transfer does not assume the data to be stationary and can be applied to randomized trials and observational studies in which treatment is confounded. Causal Transfer adjusts the effect for possible confounders and transfers the learned effect to other time series and, thereby, estimates various forms of treatment effects, such as the average treatment effect (ATE) or the conditional average treatment effect (CATE). By learning the time dynamics of the effect, Causal Transfer can also predict the treatment effect for unobserved future time points and determine the long-term consequences of treatment.
In recent years, there has been an increasing demand on efficient algorithms for large scale change point detection problems. To this end, we propose seeded binary segmentation, an approach relying on a deterministic construction of background intervals, called seeded intervals, in which single change points are searched. The final selection of change points based on the candidates from seeded intervals can be done in various ways, adapted to the problem at hand. Thus, seeded binary segmentation is easy to adapt to a wide range of change point detection problems, let that be univariate, multivariate or even high-dimensional. We consider the univariate Gaussian change in mean setup in detail. For this specific case we show that seeded binary segmentation leads to a near-linear time approach (i.e. linear up to a logarithmic factor) independent of the underlying number of change points. Furthermore, using appropriate selection methods, the methodology is shown to be asymptotically minimax optimal. While computationally more efficient, the finite sample estimation performance remains competitive compared to state of the art procedures. Moreover, we illustrate the methodology for high-dimensional settings with an inverse covariance change point detection problem where our proposal leads to massive computational gains while still exhibiting good statistical performance.
We propose a family of tests to assess the goodness-of-fit of a high-dimensional generalized linear model. Our framework is flexible and may be used to construct an omnibus test or directed against testing specific non-linearities and interaction effects, or for testing the significance of groups of variables. The methodology is based on extracting left-over signal in the residuals from an initial fit of a generalized linear model. This can be achieved by predicting this signal from the residuals using modern flexible regression or machine learning methods such as random forests or boosted trees. Under the null hypothesis that the generalized linear model is correct, no signal is left in the residuals and our test statistic has a Gaussian limiting distribution, translating to asymptotic control of type I error. Under a local alternative, we establish a guarantee on the power of the test. We illustrate the effectiveness of the methodology on simulated and real data examples by testing goodness-of-fit in logistic regression models. Software implementing the methodology is available in the R package `GRPtests'.
We introduce coroICA, confounding-robust independent component analysis, a novel ICA algorithm which decomposes linearly mixed multivariate observations into independent components that are corrupted (and rendered dependent) by hidden group-wise stationary confounding. It extends the ordinary ICA model in a theoretically sound and explicit way to incorporate group-wise (or environment-wise) confounding. We show that our proposed general noise model allows to perform ICA in settings where other noisy ICA procedures fail. Additionally, it can be used for applications with grouped data by adjusting for different stationary noise within each group. Our proposed noise model has a natural relation to causality and we explain how it can be applied in the context of causal inference. In addition to our theoretical framework, we provide an efficient estimation procedure and prove identifiability of the unmixing matrix under mild assumptions. Finally, we illustrate the performance and robustness of our method on simulated data, provide audible and visual examples, and demonstrate the applicability to real-world scenarios by experiments on publicly available Antarctic ice core data as well as two EEG data sets. We provide a scikit-learn compatible pip-installable Python package coroICA as well as R and Matlab implementations accompanied by a documentation at https://sweichwald.de/coroICA/
We provide a view on high-dimensional statistical inference for genome-wide association studies (GWAS). It is in part a review but covers also new developments for meta analysis with multiple studies and novel software in terms of an R-package hierinf. Inference and assessment of significance is based on very high-dimensional multivariate (generalized) linear models: in contrast to often used marginal approaches, this provides a step towards more causal-oriented inference.
We discuss recent work for causal inference and predictive robustness in a unifying way. The key idea relies on a notion of probabilistic invariance or stability: it opens up new insights for formulating causality as a certain risk minimization problem with a corresponding notion of robustness. The invariance itself can be estimated from general heterogeneous or perturbation data which frequently occur with nowadays data collection. The novel methodology is potentially useful in many applications, offering more robustness and better `causal-oriented' interpretation than machine learning or estimation in standard regression or classification frameworks.
Causal inference is known to be very challenging when only observational data are available. Randomized experiments are often costly and impractical and in instrumental variable regression the number of instruments has to exceed the number of causal predictors. It was recently shown in Peters et al. [2016] that causal inference for the full model is possible when data from distinct observational environments are available, exploiting that the conditional distribution of a response variable is invariant under the correct causal model. Two shortcomings of such an approach are the high computational effort for large-scale data and the assumed absence of hidden confounders. Here we show that these two shortcomings can be addressed if one is willing to make a more restrictive assumption on the type of interventions that generate different environments. Thereby, we look at a different notion of invariance, namely inner-product invariance. By avoiding a computationally cumbersome reverse-engineering approach such as in Peters et al. [2016], it allows for large-scale causal inference in linear structural equation models. We discuss identifiability conditions for the causal parameter and derive asymptotic confidence intervals in the low-dimensional setting. In the case of non-identifiability we show that the solution set of causal Dantzig has predictive guarantees under certain interventions. We derive finite-sample bounds in the high-dimensional setting and investigate its performance on simulated datasets.
We investigate the problem of inferring the causal predictors of a response $Y$ from a set of $d$ explanatory variables $(X^1,\dots,X^d)$. Classical ordinary least squares regression includes all predictors that reduce the variance of $Y$. Using only the causal predictors instead leads to models that have the advantage of remaining invariant under interventions, loosely speaking they lead to invariance across different "environments" or "heterogeneity patterns". More precisely, the conditional distribution of $Y$ given its causal predictors remains invariant for all observations. Recent work exploits such a stability to infer causal relations from data with different but known environments. We show that even without having knowledge of the environments or heterogeneity pattern, inferring causal relations is possible for time-ordered (or any other type of sequentially ordered) data. In particular, this allows detecting instantaneous causal relations in multivariate linear time series which is usually not the case for Granger causality. Besides novel methodology, we provide statistical confidence bounds and asymptotic detection results for inferring causal predictors, and present an application to monetary policy in macroeconomics.
We consider identifiability of partially linear additive structural equation models with Gaussian noise (PLSEMs) and estimation of distributionally equivalent models to a given PLSEM. Thereby, we also include robustness results for errors in the neighborhood of Gaussian distributions. Existing identifiability results in the framework of additive SEMs with Gaussian noise are limited to linear and nonlinear SEMs, which can be considered as special cases of PLSEMs with vanishing nonparametric or parametric part, respectively. We close the wide gap between these two special cases by providing a comprehensive theory of the identifiability of PLSEMs by means of (A) a graphical, (B) a transformational, (C) a functional and (D) a causal ordering characterization of PLSEMs that generate a given distribution P. In particular, the characterizations (C) and (D) answer the fundamental question to which extent nonlinear functions in additive SEMs with Gaussian noise restrict the set of potential causal models and hence influence the identifiability. On the basis of the transformational characterization (B) we provide a score-based estimation procedure that outputs the graphical representation (A) of the distribution equivalence class of a given PLSEM. We derive its (high-dimensional) consistency and demonstrate its performance on simulated datasets.
We consider the problem of structure learning for bow-free acyclic path diagrams (BAPs). BAPs can be viewed as a generalization of linear Gaussian DAG models that allow for certain hidden variables. We present a first method for this problem using a greedy score-based search algorithm. We also prove some necessary and some sufficient conditions for distributional equivalence of BAPs which are used in an algorithmic ap- proach to compute (nearly) equivalent model structures. This allows us to infer lower bounds of causal effects. We also present applications to real and simulated datasets using our publicly available R-package.
We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterisation of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set-up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterisation thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models allowing the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.
In this work we propose a framework for constructing goodness of fit tests in both low and high-dimensional linear models. We advocate applying regression methods to the scaled residuals following either an ordinary least squares or Lasso fit to the data, and using some proxy for prediction error as the final test statistic. We call this family Residual Prediction (RP) tests. We show that simulation can be used to obtain the critical values for such tests in the low-dimensional setting, and demonstrate using both theoretical results and extensive numerical studies that some form of the parametric bootstrap can do the same when the high-dimensional linear model is under consideration. We show that RP tests can be used to test for significance of groups or individual variables as special cases, and here they compare favourably with state of the art methods, but we also argue that they can be designed to test for as diverse model misspecifications as heteroscedasticity and nonlinearity.
Causal inference from observational data is an ambitious but highly relevant task, with diverse applications ranging from natural to social sciences. Within the scope of nonparametric time series, causal inference defined through interventions (cf. Pearl (2000)) is largely unexplored, although time order simplifies the problem substantially. We consider a marginal integration scheme for inferring causal effects from observational time series data, MINT-T (marginal integration in time series), which is an adaptation for time series of a method proposed by Ernest and Bühlmann (Electron. J. Statist, pp. 3155-3194, vol. 9, 2015) for the case of independent data. Our approach for stationary stochastic processes is fully nonparametric and, assuming no instantaneous effects consistently recovers the total causal effect of a single intervention with optimal one-dimensional nonparametric convergence rate $n^{-2/5}$ assuming regularity conditions and twice differentiability of a certain corresponding regression function. Therefore, MINT-T remains largely unaffected by the curse of dimensionality as long as smoothness conditions hold in higher dimensions and it is feasible for a large class of stationary time series, including nonlinear and multivariate processes. For the case with instantaneous effects, we provide a procedure which guards against false positive causal statements.
We investigate the problem of testing whether $d$ random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the $d$-dimensional joint distribution and the product of the marginals into a reproducing kernel Hilbert space and define the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between the embeddings. In the population case, the value of dHSIC is zero if and only if the $d$ variables are jointly independent, as long as the kernel is characteristic. Based on an empirical estimate of dHSIC, we define three different non-parametric hypothesis tests: a permutation test, a bootstrap test and a test based on a Gamma approximation. We prove that the permutation test achieves the significance level and that the bootstrap test achieves pointwise asymptotic significance level as well as pointwise asymptotic consistency (i.e., it is able to detect any type of fixed dependence in the large sample limit). The Gamma approximation does not come with these guarantees; however, it is computationally very fast and for small $d$, it performs well in practice. Finally, we apply the test to a problem in causal discovery.
We propose a residual and wild bootstrap methodology for individual and simultaneous inference in high-dimensional linear models with possibly non-Gaussian and heteroscedastic errors. We establish asymptotic consistency for simultaneous inference for parameters in groups $G$, where $p \gg n$, $s_0 = o(n^{1/2}/\{\log(p) \log(|G|)^{1/2}\})$ and $\log(|G|) = o(n^{1/7})$, with $p$ the number of variables, $n$ the sample size and $s_0$ denoting the sparsity. The theory is complemented by many empirical results. Our proposed procedures are implemented in the R-package hdi.