Shrinkage priors for Bayesian Substitute Confounders
Authors: Yordan P. Raykov, Hengrui Luo, Justin D. Strait, Wasiur R. KhudaBukhsh
Organizations: School of Mathematical Sciences, University of Nottingham, Nottingham, UK · Department of Statistics, Rice University, USA · Lawrence Berkeley National Laboratory, USA · Statistical Sciences Group, Los Alamos National Laboratory, USA
Abstract
Multi-cause observational studies contain information about unmeasured confounding through the dependence structure among causes. However, literal imputation of the unobserved confounder is often more complex than learning a lower-dimensional substitute score that preserves the shared assignment variation needed for stable causal adjustment. The deconfounder (Wang and Blei, 2019) and related substitute confounder methods exploit this idea, but flexible assignment models can fit the joint distribution of the causes while producing scores that over-encode the treatment vector, collapse overlap, or capture single-cause variation. We develop a Bayesian factor assignment framework for learning sparse substitute confounders that retain coarse multi-cause dependence with shrinkage priors. The theory is stated at the level of posterior concentration, factor score contraction, and overlap-preserving assignment geometry and therefore does not rely on a particular shrinkage prior. Under these conditions, the proposed regression-adjusted estimators are consistent for mean potential outcomes when the corresponding latent variable identification assumptions hold. Shrinkage priors provide a natural tool for latent structural learning: they favour low-dimensional factors supported by multiple causes, discourage effectively single-cause factors, and induce an ordering of the latent factors through progressive shrinkage. Synthetic experiments illustrate the roles of signal strength, outcome validity, and geometry-aware regularization. In an Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline analysis, sparse substitute scores recover much of the adjustment obtained by directly conditioning on invasive cerebrospinal-fluid biomarkers, while collapse diagnostics identify when fitted factors reduce to individual observed measurements.
Bayesian causal discovery is widely used for its ability to quantify epistemic uncertainty over directed acyclic graphs (DAGs) through posterior inference. However, its behaviour under latent confounding remains poorly understood, as existing work typically notes that confounding breaks identifiability without characterising how the posterior distribution over DAGs responds. In this work, we analyse posterior behaviour under latent confounding in linear Gaussian causal models, focusing on additive latent confounding between exactly two observed variables. We derive a critical correlation threshold above which the score function favours graphs with a spurious edge between the confounded variables, and show that this threshold decreases with sample size -- more data lowers the correlation required for the spurious edge to be favoured. Beyond this threshold, we characterize two distinct posterior failure regimes determined by the local structure around the confounded variables. Our findings are supported by exact posterior computations on multiple graph structures, demonstrating both the predicted failure regimes.
In causal inference, confounders are variables that influence both treatment decisions and outcomes. However, unlike as in randomized clinical trials, the treatment assignment mechanism in observational studies is not known, and it is thus unclear which covariates act as confounders. Here, we aim to generate insight for causal inference and answer: which of the observed covariates act as confounders? We introduce ConfoundingSHAP, a Shapley-based method for attributing confounding strength to individual covariates. Our contributions are twofold. First, we propose a Shapley game targeted to infer the confounding strength of the covariates. Our resulting Shapley values differ from the standard applications of SHAP explanations on causal targets, such as understanding treatment effect heterogeneity, which are ill-suited for our task. Second, as our task requires evaluating the value function over many adjustment sets, we provide a scalable TabPFN-based estimation that avoids exhaustive refitting. We demonstrate the practical value across various datasets, where ConfoundingSHAP provides informative explanations of which observed covariates drive confounding and thereby helps to provide more insight for causal inference in practice.
Marie Brockschmidt, Santo M. A. R. Thies, Maresa Schröder +5
Causal discovery aims to recover causal relationships from observed data. In various fields, exploring causal relationships among variables remains an important topic, but this task becomes challenging due to the existence of latent confounders. Ignoring such confounders can lead to false associations and incorrect edge directions. In this paper, we study the linear structural equation model with latent confounders. We propose an algorithm that iteratively identifies terminal (observed) nodes and reconstructs the directed acyclic graph of the observed variables. To do this, we recover the precision matrix of the observed variables as a sparse plus low-rank matrix: a sparse matrix captures the conditional dependencies among observed variables, while a low-rank matrix captures the combined influence of a few latent confounders. We establish that for p observed variables, r latent confounders and s edges, our procedure correctly identifies the directed causal relationship among observed variables, for n≳max{slogp,rp} samples. Experimental results validate our theoretical contributions.