arXiv subjects

Mathias Drton

Publications and source records attributed to Mathias Drton.

At least 37 records · Page 2Linked to original sources

arXiv2025

Existence of Direct Density Ratio Estimators

Many two-sample problems call for a comparison of two distributions from an exponential family. Density ratio estimation methods provide ways to solve such problems through direct estimation of the differences in natural parameters. The term direct indicates that one avoids estimating both marginal distributions. In this context, we consider the Kullback--Leibler Importance Estimation Procedure (KLIEP), which has been the subject of recent work on differential networks. Our main result shows that the existence of the KLIEP estimator is characterized by whether the average sufficient statistic for one sample belongs to the convex hull of the set of all sufficient statistics for data points in the second sample. For high-dimensional problems it is customary to regularize the KLIEP loss by adding the product of a tuning parameter and a norm of the vector of parameter differences. We show that the existence of the regularized KLIEP estimator requires the tuning parameter to be no less than the dual norm-based distance between the average sufficient statistic and the convex hull. The implications of these existence issues are explored in applications to differential network analysis.

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arXiv2025

Robust Score Matching

Proposed in Hyvärinen (2005), score matching is a parameter estimation procedure that does not require computation of distributional normalizing constants. In this work we utilize the geometric median of means to develop a robust score matching procedure that yields consistent parameter estimates in settings where the observed data has been contaminated. A special appeal of the proposed method is that it retains convexity in exponential family models. The new method is therefore particularly attractive for non-Gaussian, exponential family graphical models where evaluation of normalizing constants is intractable. Support recovery guarantees for such models when contamination is present are provided. Additionally, support recovery is studied in numerical experiments and on a precipitation dataset. We demonstrate that the proposed robust score matching estimator performs comparably to the standard score matching estimator when no contamination is present but greatly outperforms this estimator in a setting with contamination.

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arXiv2024

Algebraic Sparse Factor Analysis

Factor analysis is a statistical technique that explains correlations among observed random variables with the help of a smaller number of unobserved factors. In traditional full factor analysis, each observed variable is influenced by every factor. However, many applications exhibit interesting sparsity patterns, that is, each observed variable only depends on a subset of the factors. In this paper, we study such sparse factor analysis models from an algebro-geometric perspective. Under mild conditions on the sparsity pattern, we examine the dimension of the set of covariance matrices that corresponds to a given model. Moreover, we study algebraic relations among the covariances in sparse two-factor models. In particular, we identify cases in which a Gröbner basis for these relations can be derived via a 2-delightful term order and join of toric ideals of graphs.

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arXiv2024

Confidence Sets for Causal Orderings

Causal discovery procedures aim to deduce causal relationships among variables in a multivariate dataset. While various methods have been proposed for estimating a single causal model or a single equivalence class of models, less attention has been given to quantifying uncertainty in causal discovery in terms of confidence statements. A primary challenge in causal discovery of directed acyclic graphs is determining a causal ordering among the variables, and our work offers a framework for constructing confidence sets of causal orderings that the data do not rule out. Our methodology specifically applies to identifiable structural equation models with additive errors and is based on a residual bootstrap procedure to test the goodness-of-fit of causal orderings. We demonstrate the asymptotic validity of the confidence set constructed using this goodness-of-fit test and explain how the confidence set may be used to form sub/supersets of ancestral relationships as well as confidence intervals for causal effects that incorporate model uncertainty.

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arXiv2024

Probability Metrics for Tropical Spaces of Different Dimensions

The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning. Established comparison methods treat the standard setting that the distributions are supported in the same space. Recently, a new geometric solution has been proposed to address the more challenging problem of comparing measures in Euclidean spaces of differing dimensions. Here, we study the same problem of comparing probability distributions of different dimensions in the tropical setting, which is becoming increasingly relevant in applications involving complex data structures such as phylogenetic trees. Specifically, we construct a Wasserstein distance between measures on different tropical projective tori -- the focal metric spaces in both theory and applications of tropical geometry -- via tropical mappings between probability measures. We prove equivalence of the directionality of the maps, whether mapping from a low dimensional space to a high dimensional space or vice versa. As an important practical implication, our work provides a framework for comparing probability distributions on the spaces of phylogenetic trees with different leaf sets. We demonstrate the computational feasibility of our approach using existing optimisation techniques on both simulated and real data.

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arXiv2024

Distribution-free tests of multivariate independence based on center-outward quadrant, Spearman, Kendall, and van der Waerden statistics

Due to the lack of a canonical ordering in ${\mathbb R}^d$ for $d>1$, defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution in which multivariate ranks are obtained by transporting data points to a grid that approximates a uniform reference measure (Chernozhukov et al., 2017; Hallin, 2017; Hallin et al., 2021), thereby inducing ranks, signs, and a data-driven ordering of ${\mathbb R}^d$. We take up this new perspective to define and study multivariate analogues of the sign covariance/quadrant statistic, Spearman's rho, Kendall's tau, and van der Waerden covariances. The resulting tests of multivariate independence are fully distribution-free, hence uniformly valid irrespective of the actual (absolutely continuous) distribution of the observations. Our results provide the asymptotic distribution theory for these new test statistics, with asymptotic approximations to critical values to be used for testing independence between random vectors, as well as a power analysis of the resulting tests in an extension of the so-called (bivariate) Konijn model. This power analysis includes a multivariate Chernoff--Savage property guaranteeing that, under elliptical generalized Konijn models, the asymptotic relative efficiency of our van der Waerden tests with respect to Wilks' classical (pseudo-)Gaussian procedure is strictly larger than or equal to one, where equality is achieved under Gaussian distributions only. We similarly provide a lower bound for the asymptotic relative efficiency of our Spearman procedure with respect to Wilks' test, thus extending the classical result by Hodges and Lehmann on the asymptotic relative efficiency, in univariate location models, of Wilcoxon tests with respect to the Student ones.

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arXiv2024

Kernel-Based Differentiable Learning of Non-Parametric Directed Acyclic Graphical Models

Causal discovery amounts to learning a directed acyclic graph (DAG) that encodes a causal model. This model selection problem can be challenging due to its large combinatorial search space, particularly when dealing with non-parametric causal models. Recent research has sought to bypass the combinatorial search by reformulating causal discovery as a continuous optimization problem, employing constraints that ensure the acyclicity of the graph. In non-parametric settings, existing approaches typically rely on finite-dimensional approximations of the relationships between nodes, resulting in a score-based continuous optimization problem with a smooth acyclicity constraint. In this work, we develop an alternative approximation method by utilizing reproducing kernel Hilbert spaces (RKHS) and applying general sparsity-inducing regularization terms based on partial derivatives. Within this framework, we introduce an extended RKHS representer theorem. To enforce acyclicity, we advocate the log-determinant formulation of the acyclicity constraint and show its stability. Finally, we assess the performance of our proposed RKHS-DAGMA procedure through simulations and illustrative data analyses.

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arXiv2024

Identifying Total Causal Effects in Linear Models under Partial Homoscedasticity

A fundamental challenge of scientific research is inferring causal relations based on observed data. One commonly used approach involves utilizing structural causal models that postulate noisy functional relations among interacting variables. A directed graph naturally represents these models and reflects the underlying causal structure. However, classical identifiability results suggest that, without conducting additional experiments, this causal graph can only be identified up to a Markov equivalence class of indistinguishable models. Recent research has shown that focusing on linear relations with equal error variances can enable the identification of the causal structure from mere observational data. Nonetheless, practitioners are often primarily interested in the effects of specific interventions, rendering the complete identification of the causal structure unnecessary. In this work, we investigate the extent to which less restrictive assumptions of partial homoscedasticity are sufficient for identifying the causal effects of interest. Furthermore, we construct mathematically rigorous confidence regions for total causal effects under structure uncertainty and explore the performance gain of relying on stricter error assumptions in a simulation study.

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arXiv2024

Causal Discovery of Linear Non-Gaussian Causal Models with Unobserved Confounding

We consider linear non-Gaussian structural equation models that involve latent confounding. In this setting, the causal structure is identifiable, but, in general, it is not possible to identify the specific causal effects. Instead, a finite number of different causal effects result in the same observational distribution. Most existing algorithms for identifying these causal effects use overcomplete independent component analysis (ICA), which often suffers from convergence to local optima. Furthermore, the number of latent variables must be known a priori. To address these issues, we propose an algorithm that operates recursively rather than using overcomplete ICA. The algorithm first infers a source, estimates the effect of the source and its latent parents on their descendants, and then eliminates their influence from the data. For both source identification and effect size estimation, we use rank conditions on matrices formed from higher-order cumulants. We prove asymptotic correctness under the mild assumption that locally, the number of latent variables never exceeds the number of observed variables. Simulation studies demonstrate that our method achieves comparable performance to overcomplete ICA even though it does not know the number of latents in advance.

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arXiv2024

Conditional Independence in Stationary Diffusions

Stationary distributions of multivariate diffusion processes have recently been proposed as probabilistic models of causal systems in statistics and machine learning. Motivated by these developments, we study stationary multivariate diffusion processes with a sparsely structured drift. Our main result gives a characterization of the conditional independence relations that hold in a stationary distribution. The result draws on a graphical representation of the drift structure and pertains to conditional independence relations that hold generally as a consequence of the drift's sparsity pattern.

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arXiv2024

Causal Effect Identification in LiNGAM Models with Latent Confounders

We study the generic identifiability of causal effects in linear non-Gaussian acyclic models (LiNGAM) with latent variables. We consider the problem in two main settings: When the causal graph is known a priori, and when it is unknown. In both settings, we provide a complete graphical characterization of the identifiable direct or total causal effects among observed variables. Moreover, we propose efficient algorithms to certify the graphical conditions. Finally, we propose an adaptation of the reconstruction independent component analysis (RICA) algorithm that estimates the causal effects from the observational data given the causal graph. Experimental results show the effectiveness of the proposed method in estimating the causal effects.

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arXiv2024

Rational Maximum Likelihood Estimators of Kronecker Covariance Matrices

As is the case for many curved exponential families, the computation of maximum likelihood estimates in a multivariate normal model with a Kronecker covariance structure is typically carried out with an iterative algorithm, specifically, a block-coordinate ascent algorithm. In this article we highlight a setting, specified by a coprime relationship between the sample size and dimension of the Kronecker factors, where the likelihood equations have algebraic degree one and an explicit, easy-to-evaluate rational formula for the maximum likelihood estimator can be found. A partial converse of this result is provided that shows that outside of the aforementioned special setting and for large sample sizes, examples of data sets can be constructed for which the degree of the likelihood equations is larger than one.

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arXiv2024

Faithlessness in Gaussian graphical models

The implication problem for conditional independence (CI) asks whether the fact that a probability distribution obeys a given finite set of CI relations implies that a further CI statement also holds in this distribution. This problem has a long and fascinating history, cumulating in positive results about implications now known as the semigraphoid axioms as well as impossibility results about a general finite characterization of CI implications. Motivated by violation of faithfulness assumptions in causal discovery, we study the implication problem in the special setting where the CI relations are obtained from a directed acyclic graphical (DAG) model along with one additional CI statement. Focusing on the Gaussian case, we give a complete characterization of when such an implication is graphical by using algebraic techniques. Moreover, prompted by the relevance of strong faithfulness in statistical guarantees for causal discovery algorithms, we give a graphical solution for an approximate CI implication problem, in which we ask whether small values of one additional partial correlation entail small values for yet a further partial correlation.

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arXiv2024

Testing Many Constraints in Possibly Irregular Models Using Incomplete U-Statistics

We consider the problem of testing a null hypothesis defined by equality and inequality constraints on a statistical parameter. Testing such hypotheses can be challenging because the number of relevant constraints may be on the same order or even larger than the number of observed samples. Moreover, standard distributional approximations may be invalid due to irregularities in the null hypothesis. We propose a general testing methodology that aims to circumvent these difficulties. The constraints are estimated by incomplete U-statistics, and we derive critical values by Gaussian multiplier bootstrap. We show that the bootstrap approximation of incomplete U-statistics is valid for kernels that we call mixed degenerate when the number of combinations used to compute the incomplete U-statistic is of the same order as the sample size. It follows that our test controls type I error even in irregular settings. Furthermore, the bootstrap approximation covers high-dimensional settings making our testing strategy applicable for problems with many constraints. The methodology is applicable, in particular, when the constraints to be tested are polynomials in U-estimable parameters. As an application, we consider goodness-of-fit tests of latent tree models for multivariate data.

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arXiv2024

High-Dimensional Undirected Graphical Models for Arbitrary Mixed Data

Graphical models are an important tool in exploring relationships between variables in complex, multivariate data. Methods for learning such graphical models are well developed in the case where all variables are either continuous or discrete, including in high-dimensions. However, in many applications data span variables of different types (e.g. continuous, count, binary, ordinal, etc.), whose principled joint analysis is nontrivial. Latent Gaussian copula models, in which all variables are modeled as transformations of underlying jointly Gaussian variables, represent a useful approach. Recent advances have shown how the binary-continuous case can be tackled, but the general mixed variable type regime remains challenging. In this work, we make the simple yet useful observation that classical ideas concerning polychoric and polyserial correlations can be leveraged in a latent Gaussian copula framework. Building on this observation we propose flexible and scalable methodology for data with variables of entirely general mixed type. We study the key properties of the approaches theoretically and empirically, via extensive simulations as well an illustrative application to data from the UK Biobank concerning COVID-19 risk factors.

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arXiv2024

$\texttt{causalAssembly}$: Generating Realistic Production Data for Benchmarking Causal Discovery

Algorithms for causal discovery have recently undergone rapid advances and increasingly draw on flexible nonparametric methods to process complex data. With these advances comes a need for adequate empirical validation of the causal relationships learned by different algorithms. However, for most real data sources true causal relations remain unknown. This issue is further compounded by privacy concerns surrounding the release of suitable high-quality data. To help address these challenges, we gather a complex dataset comprising measurements from an assembly line in a manufacturing context. This line consists of numerous physical processes for which we are able to provide ground truth causal relationships on the basis of a detailed study of the underlying physics. We use the assembly line data and associated ground truth information to build a system for generation of semisynthetic manufacturing data that supports benchmarking of causal discovery methods. To accomplish this, we employ distributional random forests in order to flexibly estimate and represent conditional distributions that may be combined into joint distributions that strictly adhere to a causal model over the observed variables. The estimated conditionals and tools for data generation are made available in our Python library $\texttt{causalAssembly}$. Using the library, we showcase how to benchmark several well-known causal discovery algorithms.

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arXiv2024

Dual Likelihood for Causal Inference under Structure Uncertainty

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.

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arXiv2023

On the Lasso for Graphical Continuous Lyapunov Models

Graphical continuous Lyapunov models offer a new perspective on modeling causally interpretable dependence structure in multivariate data by treating each independent observation as a one-time cross-sectional snapshot of a temporal process. Specifically, the models assume that the observations are cross-sections of independent multivariate Ornstein-Uhlenbeck processes in equilibrium. The Gaussian equilibrium exists under a stability assumption on the drift matrix, and the equilibrium covariance matrix is determined by the continuous Lyapunov equation. Each graphical continuous Lyapunov model assumes the drift matrix to be sparse, with a support determined by a directed graph. A natural approach to model selection in this setting is to use an $\ell_1$-regularization technique that, based on a given sample covariance matrix, seeks to find a sparse approximate solution to the Lyapunov equation. We study the model selection properties of the resulting lasso technique to arrive at a consistency result. Our detailed analysis reveals that the involved irrepresentability condition is surprisingly difficult to satisfy. While this may prevent asymptotic consistency in model selection, our numerical experiments indicate that even if the theoretical requirements for consistency are not met, the lasso approach is able to recover relevant structure of the drift matrix and is robust to aspects of model misspecification.

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