Seminar on Statistics and Data Science

The availability of diverse, high-quality data has led to tremendous advances in science, technology and society at large, when analysed by means of statistical and machine learning (ML) methods. However, real-world data, in many cases, cannot be made public to the research community due to privacy restrictions, obstructing progress, especially in bio-medical research. Synthetic data can substitute the sensitive real data, and as long as they do not disclose private aspects. This has proven to be successful in training downstream ML applications. We propose TVineSynth, a vine copula based synthetic tabular data generator, which is designed to balance privacy and utility, using the vine tree structure and its truncation to do the trade-off. Contrary to synthetic data generators that achieve differential privacy (DP) by globally adding noise, TVineSynth performs a controlled approximation of the estimated data generating distribution, so that it does not suffer from poor utility of the resulting synthetic data for downstream prediction tasks. TVineSynth introduces a targeted bias into the vine copula model that, combined with the specific tree structure of the vine, causes the model to zero out privacy-leaking dependencies while relying on those that are beneficial for utility. Privacy is here measured with membership (MIA) and attribute inference attacks (AIA). Further, we theoretically justify how the construction of TVineSynth ensures AIA privacy under a natural privacy measure for continuous sensitive attributes. When compared to competitor models, with and without DP, on simulated and on real-world data, TVineSynth achieves a superior privacy-utility balance.

Recently developed quasi-Bayesian (QB) methods proposed a stimulating change of paradigm in Bayesian computation by directly constructing the Bayesian predictive distribution through recursion, removing the need for expensive computations involved in sampling the Bayesian posterior distribution. This has proved to be data-efficient for univariate predictions, however, existing constructions for higher dimensional densities are only possible by relying on restrictive assumptions on the model's multivariate structure. In this talk, we discuss a wholly different approach to extend Quasi-Bayesian prediction to high dimensions through the use of Sklar's theorem, by decomposing the predictive distribution into one-dimensional predictive marginals and a high-dimensional copula. We use the efficient recursive QB construction for the one-dimensional marginals and model the dependence using highly expressive vine copulas. Further, we tune hyperparameters using robust divergences (eg. energy score) and show that our proposed Quasi-Bayesian Vine (QB-Vine) is a fully non-parametric density estimator with an analytical form and convergence rate independent of the dimension of the data in some situations. Our experiments illustrate that the QB-Vine is appropriate for high dimensional distributions (64), needs very few samples to train (200), and outperforms state-of-the-art methods with analytical forms for density estimation and supervised tasks by a considerable margin.

Uncertainty quantification (UQ) is a crucial but challenging task in many high-dimensional regression or learning problems to increase the confidence of a given predictor. In this talk we discuss a new data-driven approach for UQ in regression that applies both to classical regression approaches such as the LASSO as well as to neural networks. One of the most notable UQ techniques is the debiased LASSO, which modifies the LASSO to allow for the construction of asymptotic confidence intervals by decomposing the estimation error into a Gaussian and an asymptotically vanishing bias component. However, in real-world problems with finite-dimensional data, the bias term is often too significant to be neglected, resulting in overly narrow confidence intervals. In this talk we will address this issue and derive a data-driven adjustment that corrects the confidence intervals for a large class of predictors by estimating the means and variances of the bias terms from training data, exploiting high-dimensional concentration phenomena. This gives rise to non-asymptotic confidence intervals, which can help avoid overestimating uncertainty in critical applications such as MRI diagnosis. Importantly, this analysis extends beyond sparse regression to data-driven predictors like neural networks, enhancing the reliability of model-based deep learning. Our findings, discussed in this talk, bridge the gap between established theory and the practical applicability of such debiased methods. This talk is based on joint work with Frederik Hoppe, Claudio Mayrink Verdun, Felix Krahmer and Holger Rauhut.

Matrix time series, which consist of matrix-valued data observed over time, are prevalent in various fields such as economics, finance, and engineering. Such matrix time series data are often observed in high dimensions. Matrix factor models are employed to reduce the dimensionality of such data, but they lack the capability to make predictions without specified dynamics in the latent factor process. To address this issue, we propose a two-component dynamic matrix factor model that extends the standard matrix factor model by incorporating a bilinear autoregressive structure for the low-dimensional latent factor process. This two-component model injects prediction capability to the matrix factor model and provides deeper insights into the dynamics of high-dimensional matrix time series. We present the estimation procedures of the model and their theoretical properties, as well as empirical analysis of the proposed procedures.

The work on parameter orthogonalisation by Cox and Reid (1987) is presented as an inducement of abstract population-level sparsity. The latter is taken as a unifying theme for the talk, in which sparsity-inducing parameterisations or data transformations are sought. Examples from some of my work are framed in this light, with emphasis on recent developments addressing the following question: for a given statistical problem, not obviously sparse in its natural formulation, can a sparsity-inducing reparametrisation be deduced? In general, the solution strategy for the problem of exact or approximate sparsity inducement appears to be context specific and may entail, for instance, solving one or more partial differential equation, or specifying a parametrised path through data-transformation or parametrisation space.

The Euler characteristic of a very affine variety encodes the algebraic complexity of solving likelihood (or scattering) equations on this variety. I will discuss this quantity for the Grassmannian with d hyperplane sections removed. I will demonstrate a combinatorial formula, and explain how to compute this Euler characteristic in practice, both symbolically and numerically. Connections to (and between) statistics and particle physics will also be discussed. This is joint work with Elia Mazzucchelli and Kexin Wang. \[ \] This talk is part of a special series of lectures on algebraic statistics.

Applying causal discovery algorithms to real-world problems can be challenging, especially when unobserved confounders may be present. In theory, constraint-based approaches like FCI are able to handle this, but implicitly rely on sparse networks to complete within a reasonable amount of time. However, in practice many relevant systems are anything but sparse. For example, in protein-protein interaction networks, there are often high-density nodes (‘hub nodes’) representing key proteins that are central to many processes in the cell. Other systems, like electric power grids and social networks, can exhibit high-clustering tendencies, leading to so-called small-world networks. In such cases, even basic constrained based algorithms like PC can get stuck in the high-density regions, failing to produce any useful output. Existing approaches to deal with this, like Anytime-FCI, were designed to interrupt the search at any stage and then produce a sound causal model output a.s.a.p., but unfortunately can require even more time to complete than just letting FCI run by itself. In this talk I will present a new approach to causal discovery in graphs with high-density regions. Based on a radically new search strategy and modified orientation rules, it builds up the causal graph on the fly, updating the model after each validated edge removal, while remaining sound throughout. It exploits the intermediate causal model to efficiently reduce number and size of the conditional independence tests, and automatically prioritizes ‘low-hanging fruit', leaving difficult (high-density) regions until last. The resulting ‘anytime-anywhere FCI+’ is a true anytime algorithm that is not only faster than its traditional counterparts, but also more flexible, and can easily be adapted to handle arbitrary edge removals as well, opening up new possibilities for e.g. targeted cause-effect recovery in large graphs.

Decision makers desire to implement decision rules that, when applied to individuals in the population of interest, yield the best possible outcomes. For example, the current focus on precision medicine reflects the search for individualized treatment decisions, adapted to a patient's characteristics. In this presentation, I will consider how to formulate, choose and estimate effects that guide individualized treatment decisions. In particular, I will introduce a class of regimes that are guaranteed to outperform conventional optimal regimes in settings with unmeasured confounding. I will further consider how to identify or bound these "superoptimal" regimes and their values. The performance of the superoptimal regimes will be illustrated in two examples from medicine and economics.

A fundamental challenge in causal induction is inferring the underlying graph structure from observational and interventional data. While traditional algorithms rely on candidate graph generation and score-based evaluation, my research takes a different approach. I developed neural causal models that leverage the flexibility and scalability of neural networks to infer causal relationships, enabling more expressive mechanisms and facilitating analysis of larger systems. Building on this foundation, I further explored causal foundation models, inspired by the success of large language models. These models are trained on massive datasets of diverse causal graphs, learning to predict causal structures from both observational and interventional data. This "black box" approach achieves remarkable performance and scalability, significantly surpassing traditional methods. This model exhibits robust generalization to new synthetic graphs, resilience under train-test distribution shifts, and achieves state-of-the-art performance on naturalistic graphs with low sample complexity. We then leverage LLMs and their metacognitive processes to causally organize skills. By labeling problems and clustering them into interpretable categories, LLMs gain the ability to categorize skills, which acts as a causal variable enabling skill-based prompting and enhancing mathematical reasoning. This integrated perspective, combining causal induction in graph structures with emergent skills in LLMs, advances our understanding of how skills function as causal variables. It offers a structured pathway to unlock complex reasoning capabilities in AI, paving the way from simple word prediction to sophisticated causal reasoning in LLMs.

Recently, the problem of predicting a response variable from a set of covariates on a data set that differs in distribution from the training data has received more attention. We propose a sequence of causal-like models from in-sample data that provide out-of-sample risk guarantees when predicting a target variable from a set of covariates. Whereas ordinary least squares provides the best in-sample risk with limited out-of-sample guarantees, causal models have the best out-of-sample guarantees by sacrificing in-sample risk performance. We introduce causal regularization by defining a trade-off between these properties. As the regularization increases, causal regularization provides estimators whose risk is more stable at the cost of increasing their overall in-sample risk. The increased risk stability is shown to result in out-of-sample risk guarantees. We provide finite sample risk bounds for all models and prove the adequacy of cross-validation for attaining these bounds.

**September 25, 2024** 09:00-09:45 Niels Richard Hansen (University of Copenhagen) 09:45-10:30 Negar Kiyavash (EPFL) break 11:00-11:45 Martin Huber (University of Fribourg) 11:45-12:30 Niklas Pfister (University of Copenhagen) lunch 14:00-14:45 Leonard Henckel (University College Dublin) 14:45-15:30 Jakob Runge (TU Dresden) **September 26, 2024** 10:00-10:45 Francesco Locatello (ISTA) 10:45-11:30 Isabel Valera (Saarland University) break 11:45-12:30 Sara Magliacane (University of Amsterdam) lunch 14:00-14:45 Qingyuan Zhao (University of Cambridge) 14:45-15:30 Jalal Etesami (Technical University of Munich) See https://collab.dvb.bayern/display/TUMmathstat/Miniworkshop+on+Causal+Inference+2024 for more details.

We consider the problem of making nonparametric inference in multi-dimensional diffusion models from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition densities as solutions of the associated heat (Fokker-Planck) equation. Employing optimal regularity results from the theory of parabolic PDEs, we prove a novel characterisation for the gradient of the likelihood. Using these developments, for the nonlinear inverse problem of recovering the diffusivity (in divergence form models), we then show that the numerical evaluation of the likelihood and its gradient can be reduced to standard elliptic eigenvalue problems, solvable by powerful finite element methods. This enables the efficient implementation of a large class of statistical algorithms, including (i) preconditioned Crank-Nicolson and Langevin-type methods for posterior sampling, and (ii) gradient-based descent optimisation schemes to compute maximum likelihood and maximum-a-posteriori estimates. We showcase the effectiveness of these methods via extensive simulation studies in a nonparametric Bayesian model with Gaussian process priors. Interestingly, the optimisation schemes provided satisfactory numerical recovery while exhibiting rapid convergence towards stationary points despite the problem nonlinearity; thus our approach may lead to significant computational speed-ups.