ICML 2025

Latent Variable Causal Discovery under Selection Bias
Haoyue Dai 1 2 Yiwen Qiu 1 Ignavier Ng 1 Xinshuai Dong 1 Peter Spirtes 1 Kun Zhang 1 2
Abstract
Addressing selection bias in latent variable causal
discovery is important yet underexplored, largely
due to a lack of suitable statistical tools: While
various tools beyond basic conditional indepen-
dencies have been developed to handle latent vari-
ables, none have been adapted for selection bias.
We make an attempt by studying rank constraints,
which, as a generalization to conditional inde-
pendence constraints, exploits the ranks of co-
variance submatrices in linear Gaussian models.
We show that although selection can significantly
complicate the joint distribution, interestingly, the
ranks in the biased covariance matrices still pre-
serve meaningful information about both causal
structures and selection mechanisms. We provide
a graph-theoretic characterization of such rank
constraints. Using this tool, we demonstrate that
the one-factor model, a classical latent variable
model, can be identified under selection bias. Sim-
ulations and real-world experiments confirm the
effectiveness of using our rank constraints.
1
Introduction
At the core of understanding complex systems lies causal
discovery, the identification of causal relations from obser-
vational data (Spirtes et al., 2000; Pearl, 2009). In many
real-world scenarios, the variables of interest are latent
constructs that cannot be directly observed or quantized,
while the observed variables serve merely as indirect
measurements. For instance, in psychological or political-
economic surveys, measured responses serve as proxies for
latent personality traits or political orientations. Recovering
the causal structure among these latent variables—referred
to as latent variable causal discovery—is essential for un-
derstanding and reasoning, yet remains a challenging task.
Furthermore, a typical assumption in causal discovery,
1Carnegie Mellon University 2Moham...