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Publications and Preprints

J. Bodik, Y. Huang and B. Yu (2025, preprint, submitted to JRSSb)
Cross-World Assumption and Refining Prediction Intervals for Individual Treatment Effects

Abstract: While average treatment effects (ATE) and conditional average treatment effects (CATE) provide valuable population- and subgroup-level summaries, they fail to capture uncertainty at the individual level. For high-stakes decision-making, individual treatment effect (ITE) estimates must be accompanied by valid prediction intervals that reflect heterogeneity and unit-specific uncertainty. However, the fundamental unidentifiability of ITEs limits the ability to derive precise and reliable individual-level uncertainty estimates. To address this challenge, we investigate the role of a cross-world correlation parameter, \(\rho(x) = \text{cor}\big(Y(1), Y(0) \mid X = x\big)\), which describes the dependence between potential outcomes, given covariates, in the Neyman-Rubin super-population model with i.i.d. units. Although \(\rho\) is fundamentally unidentifiable, we argue that in most real-world applications, it is possible to impose reasonable and interpretable bounds informed by domain-expert knowledge. Given \(\rho\), we design prediction intervals for ITE, achieving more stable and accurate coverage with substantially shorter widths; often less than 1/3 of those from competing methods. The resulting intervals satisfy coverage guarantees \(P\big(Y(1) - Y(0) \in C_{ITE}(X)\big) \geq 1 - \alpha\) and are asymptotically optimal under Gaussian assumptions. We provide strong theoretical and empirical arguments that cross-world assumptions can make individual uncertainty quantification both practically informative and statistically valid.

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J. Bodik (2025, preprint, submitted to CleaR)
Counterfactual prediction under cross-world dependence

Abstract: We study the problem of estimating the expected counterfactual outcome for an individual with covariates \(x\) and observed outcome \(y\), defined as \(\mu(x,y) = \mathbb{E}[Y(1) \mid X = x, Y(0) = y]\), and constructing valid prediction intervals under the Neyman–Rubin superpopulation model with i.i.d. units. This quantity is generally unidentified without additional assumptions. To link the observed and unobserved potential outcomes, we work with a cross-world correlation function \(\rho(x) = \operatorname{cor}(Y(1), Y(0) \mid X = x)\) that quantifies their dependence given the covariates. Plausible bounds on \(\rho(x)\), often informed by domain knowledge, enable a principled approach to this otherwise unidentified problem. Given \(\rho\), we develop a consistent estimator \(\hat{\mu}_{\rho}(x,y)\) and prediction intervals \(C_{\rho}(x,y)\) that satisfy \(P[Y(1) \in C_{\rho}(X,Y(0))] \geq 1 - \alpha\) under standard causal assumptions. Almost all existing methods correspond to either the case \(\rho = 0\) (ignoring the factual outcome), or \(\rho = 1\) (constant treatment effects). We show that interpolating between these cases via cross-world dependence yields estimators that are theoretically optimal under (asymptotic) Gaussian assumptions. In practice, this leads to substantial empirical improvements across a wide range of scenarios.

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I. Azizi^✶, J. Bodik^✶, J. Heiss^✶, and B. Yu (2025, preprint, submitted to ICLR)
^✶Equal contribution
CLEAR: Calibrated Learning for Epistemic and Aleatoric Risk

Abstract: Accurate uncertainty quantification is critical for reliable predictive modeling, especially in regression tasks. Existing methods typically address either aleatoric uncertainty from measurement noise or epistemic uncertainty from limited data, but not necessarily both in a balanced way. We propose CLEAR, a calibration method with two distinct parameters, and , to combine the two uncertainty components for improved conditional coverage. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for aleatoric uncertainty and (ii) ensembles drawn from the Predictability-Computability-Stability (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves an average improvement of 28.2% and 17.4% in the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. This improvement can be particularly evident in scenarios dominated by either high epistemic or high aleatoric uncertainty.

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J. Bodik and O. Pasche (2024, preprint, submitted to Bernoulli)
Granger causality in extremes

Abstract: We introduce a rigorous mathematical framework for Granger causality in extremes, designed to identify causal links from extreme events in time series. Granger causality plays a pivotal role in uncovering directional relationships among time-varying variables. While this notion gains heightened importance during extreme and highly volatile periods, state-of-the-art methods primarily focus on causality within the body of the distribution, often overlooking causal mechanisms that manifest only during extreme events. Our framework is designed to infer causality mainly from extreme events by leveraging the causal tail coefficient. We establish equivalences between causality in extremes and other causal concepts, including (classical) Granger causality, Sims causality, and structural causality. We prove other key properties of Granger causality in extremes and show that the framework is especially helpful under the presence of hidden confounders. We also propose a novel inference method for detecting the presence of Granger causality in extremes from data. Our method is model-free, can handle non-linear and high-dimensional time series, outperforms current state-of-the-art methods in all considered setups, both in performance and speed, and was found to uncover coherent effects when applied to financial and extreme weather observations.

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J. Bodik and V. Chavez-Demoulin (2025, Journal of Machine Learning Research)
Identifiability of causal graphs under nonadditive conditionally parametric causal models

Abstract: Causal discovery from observational data is a very challenging, often impossible, task. However, estimating the causal structure is possible under certain assumptions on the data-generating process. Many commonly used methods rely on the additivity of the noise in the structural equation models. Additivity implies that the variance or the tail of the effect, given the causes, is invariant; the cause only affects the mean. However, the tail or other characteristics of the random variable can provide different information about the causal structure. Such cases have received only very little attention in the literature. It has been shown that the causal graph is identifiable under different models, such as linear non-Gaussian, post-nonlinear, or quadratic variance functional models. We introduce a new class of models called the Conditional Parametric Causal Models (CPCM), where the cause affects the effect in some of the characteristics of interest. We use sufficient statistics to show the identifiability of the CPCM models in the exponential family of conditional distributions. We also propose an algorithm for estimating the causal structure from a random sample under CPCM. Its empirical properties are studied for various data sets, including an application on the expenditure behavior of residents of the Philippines.

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J. Bodik and V. Chavez-Demoulin (2025, Biometrika)
Structural restrictions in local causal discovery: identifying direct causes of a target variable

Abstract: We consider the problem of learning a set of direct causes of a target variable from an observational joint distribution. Learning directed acyclic graphs (DAGs) that represent the causal structure is a fundamental problem in science. Several results are known when the full DAG is identifiable from the distribution, such as assuming a nonlinear Gaussian data-generating process. Often, we are only interested in identifying the direct causes of one target variable (local causal structure), not the full DAG. In this paper, we discuss different assumptions for the data-generating process of the target variable under which the set of direct causes is identifiable from the distribution. While doing so, we put essentially no assumptions on the variables other than the target variable. In addition to the novel identifiability results, we provide two practical algorithms for estimating the direct causes from a finite random sample and show their effectiveness on several benchmark data-sets. We apply this framework to learn direct causes of the reduction in the fertility rate in different countries.

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J. Bodik (2024, Mathematics)
Extreme Treatment Effect: Extrapolating Causal Effects Into Extreme Treatment Domain

Abstract: The potential outcomes framework serves as a fundamental tool for quantifying the causal effects. When the treatment variable (exposure) is continuous, one is typically interested in the estimation of the effect curve (also called the average dose-response function), denoted as mu(t). In this work, we explore the ``extreme causal effect,’’ where our focus lies in determining the impact of an extreme level of treatment, potentially beyond the range of observed values–that is, estimating mu(t) for very large t. Our framework is grounded in the field of statistics known as extreme value theory. We establish the foundation for our approach, outlining key assumptions that enable the estimation of the extremal causal effect. Additionally, we present a novel and consistent estimation procedure that utilizes extreme value theory in order to potentially reduce the dimension of the confounders to at most 3. In practical applications, our framework proves valuable when assessing the effects of scenarios such as drug overdoses, extreme river discharges, or extremely high temperatures on a variable of interest

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J. Bodik, Z. Pawlas, M. Paluš (2024, Extremes)
Causality in extremes of time series.

Abstract: Consider two stationary time series with heavy-tailed marginal distributions. We aim to detect whether they have a causal relation, that is, if a change in one of them causes a change in the other. Usual methods for causality detection are not well suited if the causal mechanisms only manifest themselves in extremes. This paper aims to detect the causal relations in extremes between time series. We define the so-called causal tail coefficient for time series, which, under some assumptions, correctly detects the asymmetrical causal relations between extremes of the time series. The advantage is that this method works even if nonlinear relations and common ancestors are present. Moreover, we mention how our method can help detect a time delay between the two time series. We describe some of its asymptotic properties and show how it performs on some simulations. Finally, we show how this method works on space-weather and hydro-meteorological data sets.

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J. Bodik, L. Mhalla, V. Chavez-Demoulin (2022 - project permanently on hold)
Detecting causal covariates for extreme dependence structures

Abstract: Determining the causes of extreme events is a fundamental question in many scientific fields. An important aspect when modelling multivariate extremes is the tail dependence. In application, the extreme dependence structure may significantly depend on covariates. As for the general case of modelling including covariates, only some of the covariates are causal. In this paper, we propose a methodology to discover the causal covariates explaining the tail dependence structure between two variables. The proposed methodology for discovering causal variables is based on comparing observations from different environments or perturbations. It is a desired methodology for predicting extremal behaviour in a new, unobserved environment. The methodology is applied to a dataset of NO2 concentration in the UK. Extreme NO2 levels can cause severe health problems, and understanding the behaviour of concurrent severe levels is an important question. We focus on revealing causal predictors for the dependence between extreme NO2 observations at different sites.
J. Bodik, Z. Pawlas, M. Paluš (2021, Master’s thesis; prize 10,000 Kč for best thesis)
Detection of causality in time series using extreme values.

Abstract: In this thesis we deal with the following problem: Let us have two stationary time series with heavy-tailed marginal distributions. We want to detect whether they have a causal relation, i.e. if a change in one of them causes a change in the other. The question of distinguishing between causality and correlation is essential in many different science fields. Usual methods for causality detection are not well suited if the causal mechanisms only manifest themselves in extremes. In this thesis, we propose a new method that can help us in such a non-traditional case distinguish between correlation and causality. We define the so-called causal tail coefficient for time series, which, under some assumptions, correctly detects the asymmetrical causal relations between different time series. We will rigorously prove this claim, and we also propose a method on how to statistically estimate the causal tail coefficient from a finite number of data. The advantage is that this method works even if nonlinear relations and common ancestors are present. Moreover, we will mention how our method can help detect a time delay between the two time series. We will show how our method performs on some simulations. Finally, we will show on a real data set how this method works, discussing a cause of electromagnetic storms.

Reviewing
Journal of the Royal Statistical Society, Series B (JRSSb), Statistics & Probability Letters, Behaviormetrika, REVSTAT - Statistical Journal, Data Mining and Knowledge Discovery

Selected conferences and talks

Invited talks

Online Causal Inference Seminar (The largest international seminar on causal inference, Online, 3.2.2026)
SANKEN+RIKEN Data science seminar (Seminar on causality between SANKEN and RIKEN organisations, University of Osaka, Osaka, Japan, 26.12.2025)
CHUV BDSC seminar (Clinical Data Science Group seminar at CHUV, Lausanne, Switzerland, 29.9.2025)
UAI 2024 (Workshop on Causal Inference in Time Series, Barcelona, Spain, 15.7.2024–19.7.2024)

Other conferences and talks

SANKEN seminar (Data science lab at University of Osaka, Osaka, Japan, 16.12.2025)
Data Driven seminar (Weekly research discussion seminar around data-driven decisions and inference, Stanford, California, 7.11.2024)
Causal Seminar, PhD Students Seminar, and Bin Yu Group Seminar (Speaker, UC Berkeley, California, autumn 2024)
Causal Science Center Conference 2024 (Presented + poster, Stanford, California, 11.10.2024)
EGU24 (Poster, Vienna, Austria, 14.4.2024–19.4.2024)
Causality in Extremes workshop (Poster, Geneva, Switzerland, 12.2.2024–14.2.2024)
CMStatistics 2023 (Presented, HTW Berlin, Germany, 15.12.2023–19.12.2023)
EUROCIM: European Causal Inference Meeting (Presented, University of Oslo, Norway, 18.4.2023–22.4.2023)
CUSO: Statistics and Applied Probability (Participant, Les Diablerets, Switzerland: 5.2.2023–8.2.2023; 4.9.2022–7.9.2022; 6.2.2022–9.2.2022; 12.9.2021–15.9.2021)
BIRS: Combining Causal Inference and Extreme Value Theory in the Study of Climate Extremes and their Causes (Presented, Zoom / Kelowna, Canada, 26.6.2022–1.7.2022)
ROBUST (Presented, Prague, Czech Republic, 12.6.2022–17.6.2022)
CMStatistics 2021 (Presented, Zoom / London, UK, 18.12.2021–20.12.2021)
VALPRED workshop (Presented, Aussois, France, 4.10.2021–7.10.2021)
EVA: Extreme Value Analysis (Presented, Zoom / Edinburgh, UK, 5.6.2021–9.6.2021)

Keywords: Mathematical statistics; Causal inference; Statistical inference; Probability theory; Hypothesis testing; Estimation theory; Confidence intervals; Statistical models; Causality; Counterfactuals; Treatment effects; Observational studies; Randomized experiments; Potential outcomes; Causal diagrams; Identification; Confounding; Selection bias; Propensity scores; Instrumental variables; Mediation analysis; Sensitivity analysis; Bayesian statistics; Nonparametric methods; Structural equation modeling; Granger causality; Directed acyclic graphs (DAGs); Structural causal models; Regression analysis; Time series analysis; Conditional probability; Markov chains; Estimator bias; Maximum likelihood estimation; Resampling methods; Cross-validation; Model selection; Robust statistics; Multivariate analysis; Experimental design; Statistical power; Sequential analysis; Missing data; Latent variable models; Bayesian networks; Marginalization; Causal effect heterogeneity; Principal component analysis; Survival analysis; Time-to-event data; Nonlinear regression; Longitudinal data analysis; Factor analysis; Structural equation models; Bootstrap methods; Spatial statistics; Cluster analysis; Decision trees; Dimensionality reduction; Bayesian hierarchical models; Statistical learning; Machine learning; Classification methods; Regression analysis; Propensity score matching; Network analysis; Granger causality testing; Observational data analysis; Quasi-experimental designs; Robust causal inference; Counterfactual estimation; Time series modeling; Structural equation modeling; Instrumental variable regression; Causal mediation analysis; Nonignorable missing data; Multiple imputation; Factorial designs; Cluster randomized trials; Survival analysis; Propensity-based weighting; Propensity score calibration; Sensitivity analysis; Nonparametric causal inference; Random forests; Causal discovery algorithms; Generalized linear models; Bayesian model averaging; Longitudinal causal inference; Structural causal models; Neural networks; Deep learning; Artificial neural networks; Feedforward networks; Backpropagation; Activation functions; Convolutional neural networks (CNN); Recurrent neural networks (RNN); Long Short-Term Memory (LSTM); Gated Recurrent Units (GRU); Autoencoders; Generative adversarial networks (GAN); Transfer learning; Fine-tuning; Dropout regularization; Batch normalization; Gradient descent; Stochastic gradient descent (SGD); Mini-batch gradient descent; Learning rate; Momentum; Adaptive learning rate methods (e.g., Adam, RMSprop); Hyperparameters; Overfitting; Underfitting; Model capacity; Weight initialization; Convergence criteria; Loss functions; Optimizers.