I-FLOP: Fast Learning of Order and Parents from Interventional Data

We extend the FLOP (fast learning of order and parents) algorithm recently proposed by Wienöbst et al. (2026) from observational to interventional data. In particular, we use the interventional BIC score of Hauser and Bühlmann (2012), adapting it to be used with the iterative Cholesky-based score updates that are partly responsible for FLOP's speed. We show that, in the sample limit, I-FLOP recovers a DAG in the same interventional Markov equivalence class as the data-generating DAG. We compare I-FLOP to existing causal structure learning algorithms on real and simulated interventional data, where it performs favorably in terms of both performance and run time.

Evidence-Type Competition: When Can Interventional Data Teach Language Models Causal Direction?

Interventional data is widely regarded as the gold standard for teaching models causal reasoning. We test this assumption in a fully controlled synthetic environment pitting observational correlation against causal effect, and find it fails instructively. In Simpson's-paradox worlds, where the two have systematically opposite signs, increasing the fraction of interventional samples in pretraining does not improve causal direction: the magnitude of the model's do()-response grows monotonically, yet its sign is copied from the observational context. What governs whether interventional evidence is used is not the training mixture but the evidence type present in the context at inference time. Under an identical training recipe, a purely observational context induces systematic sign reversal in 29/50 worlds, a mixed context in 19/50, while aligned interventional probes alone yield 41/50 correct. Erasing observational evidence from the context immediately releases the suppressed causal interpolation ability (ratio_true = +0.56); a four-state content manipulation shows the switch is content-mediated and graded. The suppression is stable across training seeds (11/11 strong reversals persist on a matched-protocol second seed) and robust as a rate at 0.93B parameters (31.8% vs. 6% reversals in the matched probe-only arm), even as absolute gains shrink four-fold. An external audit on CLadder exposes a learned positive-effect prior with a two-layer structure: sign-randomized retraining removes it in-distribution but not out-of-distribution. We summarize: the capability lives in the weights; the switch lives in the context, and activation patching localizes the switch to the middle layers' observational rows. We further quantify the sampling noise floor of probe-based causal evaluation and an evidence-averaging protocol that cuts sign errors from 26% to 9%.

Latent Confounded Causal Discovery via Lie Bracket Geometry

We study causal discovery from observational and interventional regimes when latent variables may affect the measured system. Our first algorithm, BRIDGE (Bracket Residuals for Interventional Discovery and Geometric Estimation), combines a density-ratio or transport engine with a high-recall geometric screen and passes the retained arrows to a score-based or differentiable discovery method. The main formulation and experiments use known single-node intervention targets; in that regime the screen is designed to retain candidate directed effects, while a downstream learner determines the final graph or equivalence-class representation. Our second algorithm, Spectral Kernel Flow Matching (SKFM), amortizes the response fields, summarizes residual nonclosure by a spectral visible-footprint subspace, and applies an order-dependent graph extractor. Direct extraction succeeds on calibrated chains and selected motifs, but is unstable on harder random DAGs when the order must be learned. On ten-node nonlinear random DAGs, the more reliable hybrid role of the geometry is as a candidate generator: calibrated SKFM/Bridge fields followed by local BIC scoring achieve mean directed $F_1\simeq0.86$. Sachs protein signaling provides a real-data stress test and supports a diagnostic, not fully identified, interpretation. The contribution is therefore a practical interventional screening pipeline, explicit guarantees for screen retention and residual-footprint rank under stated assumptions, and a falsifiable account of the boundary between geometric diagnostics and causal identification.

Tensor-based second-order causal discovery

Causal discovery seeks to uncover the causal dependencies among variables. For this purpose, we propose an algorithm called Tensor-based Second-order Causal Discovery (TSCD). Its input is a tensor obtained from the covariance matrices of observational and interventional data. Assuming the causal dependencies follow a linear structural equation model on a directed acyclic graph (DAG), TSCD outputs the DAG and the functions on its edges, requiring only that the noise variables are uncorrelated. We also implement a version of the approach for nonlinear models. Our focus on second-order statistics (via the covariance matrices) is motivated by their statistical and computational efficiency relative to higher-order moments, their identifiability relative to first-order statistics, and that they work regardless of whether the variables are Gaussian. We show that TSCD has identifiable causal order and parameters from a number of interventions that is logarithmic in the number of variables. Experiments show that TSCD is robust to noise, competitive with existing methods, and scales to hundreds of variables.