Causal Discovery with Mixed Latent Confounding via Precision Decomposition

We study causal discovery from observational data in linear Gaussian systems affected by \emph{mixed latent confounding}, where some unobserved factors act broadly across many variables while others influence only small subsets. This setting is common in practice and poses a challenge for existing methods: differentiable and score-based DAG learners can misinterpret global latent effects as causal edges, while latent-variable graphical models recover only undirected structure. We propose \textsc{DCL-DECOR}, a modular, precision-led pipeline that separates these roles. The method first isolates pervasive latent effects by decomposing the observed precision matrix into a structured component and a low-rank component. The structured component corresponds to the conditional distribution after accounting for pervasive confounders and retains only local dependence induced by the causal graph and localized confounding. A correlated-noise DAG learner is then applied to this deconfounded representation to recover directed edges while modeling remaining structured error correlations, followed by a simple reconciliation step to enforce bow-freeness. We provide identifiability results that characterize the recoverable causal target under mixed confounding and show how the overall problem reduces to well-studied subproblems with modular guarantees. Synthetic experiments that vary the strength and dimensionality of pervasive confounding demonstrate consistent improvements in directed edge recovery over applying correlated-noise DAG learning directly to the confounded data.

DAG DECORation: Continuous Optimization for Structure Learning under Hidden Confounding

We study structure learning for linear Gaussian SEMs in the presence of latent confounding. Existing continuous methods excel when errors are independent, while deconfounding-first pipelines rely on pervasive factor structure or nonlinearity. We propose \textsc{DECOR}, a single likelihood-based and fully differentiable estimator that jointly learns a DAG and a correlated noise model. Our theory gives simple sufficient conditions for global parameter identifiability: if the mixed graph is bow free and the noise covariance has a uniform eigenvalue margin, then the map from $(\B,\OmegaMat)$ to the observational covariance is injective, so both the directed structure and the noise are uniquely determined. The estimator alternates a smooth-acyclic graph update with a convex noise update and can include a light bow complementarity penalty or a post hoc reconciliation step. On synthetic benchmarks that vary confounding density, graph density, latent rank, and dimension with $n<p$, \textsc{DECOR} matches or outperforms strong baselines and is especially robust when confounding is non-pervasive, while remaining competitive under pervasiveness.

Sample Selection Bias in Evaluation of Prediction Performance of Causal Models

Causal models are notoriously difficult to validate because they make untestable assumptions regarding confounding. New scientific experiments offer the possibility of evaluating causal models using prediction performance. Prediction performance measures are typically robust to violations in causal assumptions. However prediction performance does depend on the selection of training and test sets. In particular biased training sets can lead to optimistic assessments of model performance. In this work, we revisit the prediction performance of several recently proposed causal models tested on a genetic perturbation data set of Kemmeren [Kemmeren et al., 2014]. We find that sample selection bias is likely a key driver of model performance. We propose using a less-biased evaluation set for assessing prediction performance on Kemmeren and compare models on this new set. In this setting, the causal model tested have similar performance to standard association based estimators such as Lasso. Finally we compare the performance of causal estimators in simulation studies which reproduce the Kemmeren structure of genetic knockout experiments but without any sample selection bias. These results provide an improved understanding of the performance of several causal models and offer guidance on how future studies should use Kemmeren.

A Flexible Procedure for Mixture Proportion Estimation in Positive--Unlabeled Learning

Positive--unlabeled (PU) learning considers two samples, a positive set P with observations from only one class and an unlabeled set U with observations from two classes. The goal is to classify observations in U. Class mixture proportion estimation (MPE) in U is a key step in PU learning. Blanchard et al. [2010] showed that MPE in PU learning is a generalization of the problem of estimating the proportion of true null hypotheses in multiple testing problems. Motivated by this idea, we propose reducing the problem to one dimension via construction of a probabilistic classifier trained on the P and U data sets followed by application of a one--dimensional mixture proportion method from the multiple testing literature to the observation class probabilities. The flexibility of this framework lies in the freedom to choose the classifier and the one--dimensional MPE method. We prove consistency of two mixture proportion estimators using bounds from empirical process theory, develop tuning parameter free implementations, and demonstrate that they have competitive performance on simulated waveform data and a protein signaling problem.