Seminar on Statistics and Data Science

We present FLOP (Fast Learning of Order and Parents), a score-based causal discovery algorithm for linear models. It pairs fast parent selection with iterative Cholesky-based score updates, cutting run-times over prior algorithms. This makes it feasible to fully embrace discrete search, enabling iterated local search with principled order initialization to find graphs with scores at or close to the global optimum. The resulting structures are highly accurate across benchmarks, with near-perfect recovery in standard settings. This performance calls for revisiting discrete search over graphs as a reasonable approach to causal discovery.

Continuous-time stochastic systems are often used to model causal relationships between variables that evolve over time. In many applications, however, the full time evolution is not available. Instead, we only observe the system after it has reached stationarity. This leads to the question of which causal information can be recovered from such stationary data. In this talk, I will consider this question for linear stochastic differential equations when the causal structure is known. I will explain why identifying the exact strength of a direct causal effect is generally not possible without additional assumptions. Instead, I will focus on whether a direct causal effect is positive or negative. This leads to the notion of edge-sign identifiability, which asks when the sign of a direct causal effect is uniquely determined by the stationary covariance matrix. I will present criteria for determining whether an edge sign is identifiable, non-identifiable, or partially identifiable, and illustrate these criteria using examples from both classical causal structures and cyclic systems.

Functional data analysis (FDA) provides statistical methods for analyzing samples of time-continuous stochastic processes. Measurements often arise in the form of sensor data for a scientific variable. The practical problem of irregular sensor disruptions has fostered interest in analyzing partially observed random functions. To allow statistical analysis, we develop the first inference methods for dependent, partially observed functional time series. Mathematically, we model data on the space of bounded functions equipped with the supremum norm. This allows simultaneous inference across the entire functional domain, including simultaneous confidence bands -- something existing Hilbert-space-based methods cannot provide. To study non-stationary trends along the time series, we extend state-of-the-art multiscale inference methods (originally developed for scalar data) to partially observed functions. The key application of the latter methods is testing for excessive pollution levels in inner cities. Interestingly, our results also improve on existing results for fully observed functional time series by avoiding a functional CLT.

The quantity of interest in the classical Cramér–Rao theory of unbiased estimation (i.e., the Cramér–Rao lower bound, exact efficiency in exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently sampled data set from the same distribution. In this paper, we study a different quantity that captures the instability of an estimator when its value is compared to that obtained under an infinitesimal additive perturbation of the original data set; we refer to this as the sensitivity of an estimator. The resulting theory of sensitivity is based on Wasserstein geometry in much the same way that the classical theory of variance is based on Fisher–Rao (equivalently, Hellinger) geometry. This perspective yields several results paralleling the classical case: a Wasserstein–Cramér–Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models admitting unbiased estimators that attain this bound exactly, and a guarantee that Wasserstein projection estimators achieve the bound asymptotically. We illustrate the theory through a range of statistical examples, in some cases revealing new optimality properties of existing estimators and in others introducing new ones. This is joint work with Nicolas Garcia Trillos (University of Wisconsin) and Adam Jaffe (Columbia), based on the paper: https://arxiv.org/pdf/2511.07414.

Conditional-independence-based discovery uses statistical tests to identify a graphical model that represents the independence structure of variables in a dataset. These tests, however, can be unreliable, and algorithms are sensitive to errors and violated assumptions. Often, there are tests that were not used in the construction of the graph. In this talk, we will see that these redundant tests have the potential to detect or sometimes correct errors in the learned model. But we will further see that not all tests contain this additional information and that such redundant tests have to be applied with care. Precisely, we argue that the conditional (in)dependence statements that hold for every probability distribution are unlikely to detect and correct errors - in contrast to those that follow only from graphical assumptions.

Generative modeling builds on and substantially extends the classical idea of generating synthetic data from observed samples. In this talk, I will show that this principle provides a natural and theoretically well-founded foundation for bootstrap inference. The resulting method yields statistically valid confidence intervals for both regular and irregular estimators, including settings in which Efron’s bootstrap fails. From this perspective, the generative-modeling bootstrap could be viewed as a modern extension of the smoothed bootstrap: it has the potential to mitigate the curse of dimensionality and remain effective in challenging regimes where estimators may not be root-n consistent or admit a Gaussian limit.

In network science, as in other applied fields, answering key questions often requires distinguishing causal mechanisms from context-dependent associations. One formalization of causality uses invariance: a relationship between a target variable and a set of covariates is causal if it remains stable across diverse environments or interventions. Originally developed for linear models, this approach has been extended to more flexible frameworks, including generalized linear models, where causality is expressed via Pearson risk invariance. In this talk, we extend causal invariance to dynamic networks, focusing on Relational Event Models (REMs), which describe instantaneous interactions over time between social actors. Because the target —the dynamic network— and potential causal drivers are stochastic processes, dependences among them are characterized using conditional local independence. By exploiting a connection between REMs and logistic regression, we extend Pearson risk invariance to this dynamic setting, producing a causal discovery algorithm that only requires data from a single observational environment. We show how this approach is able to identify, and distinguish, important causal mechanisms in network science, such as social influence and homophily. This is joint work with Melania Lembo and Ernst Wit (USI).

Contemporary data analysis pipelines often involve the use and reuse of data. For instance, a scientist may explore a dataset to select an interesting hypothesis, and then wish to test this hypothesis with the same data. From a statistical perspective, this double use of data is highly problematic: it induces dependence between the hypothesis generation and testing stages, which complicates inference. Failure to account for this dependence renders classical inference techniques invalid. I will present "data thinning", a set of strategies for obtaining independent training and test sets so that the former can be used to select a hypothesis, and the latter to test it. Data thinning enables valid selective inference in settings for which no solutions were previously available. However, it is also restrictive, in the sense that it requires strong distributional assumptions. Therefore, I will also present two strategies inspired by data thinning that enable valid post-selection inference without such assumptions. One strategy considers thinning summary statistics of the data, rather than the data itself, in order to take advantage of asymptotic properties of the summary statistics. The second strategy involves generating training and test sets that are not independent, and then orthogonalizing the latter with respect to the former in order to conduct valid inference.

Mercer's celebrated theorem is refined and extended by introducing a novel class of higher-order kernel operators that includes the common integral operator only as a special case. These operators genuinely take into account information encoded in the (weak) derivatives of a kernel, and their natural domains are Sobolev spaces of order k over some bounded d-dimensional space. domain, where k depends on the order of (weak) differentiability. The spectral decomposition of such higher-order kernel operators leads to Mercer-type expansions, which are optimal in terms of the Sobolev norm and, if k>d, also converge uniformly without requiring the kernel to be positive definite. Nuclearity of higher order kernel operators is confirmed for positive definite kernels, and a major refinement of Mercer's theorem is obtained that implies trace formulas and a simple rate for the uniform convergence (including derivatives) in terms of the eigenvalues. A further immediate consequence is novel spectral representations of RKHS's. Finally, applied to the covariance kernel of a (weakly) differentiable stochastic process, these refinements also yield novel Karhunen-Loève-type expansions allowing for simultaneous approximations of the process and its (weak) derivatives in a mean-square-optimal sense.

ICA-based causal discovery methods such as LiNGAM have been highly successful under the assumption that noise variables become independent after an appropriate causal ordering. However, this assumption is often violated in the presence of confounding. \[ \] In this talk, I present a Bayesian and information-theoretic formulation of ICA for causal order estimation that explicitly allows for confounding. Rather than enforcing independence, we quantify residual dependence among noise variables using multivariate mutual information and evaluate causal orders via Bayesian marginal likelihoods. \[ \] This approach provides a principled ranking of causal orders under confounding and recovers classical LiNGAM-type methods as special cases when confounding is absent. I will focus on the conceptual framework and discuss connections to existing ICA-based methods, as well as open questions.