# Causal Dantzig: fast inference in linear structural equation models with hidden variables under additive interventions

arXiv: https://arxiv.org/abs/1706.06159 · HTML: https://mlsanity.com/paper/1706.06159/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2017.

## Built on by (most cited first, 10 of 10)

- [Environment Invariant Linear Least Squares](https://mlsanity.com/paper/2303.03092/) (2023)
- [Optimization-based Causal Estimation from Heterogenous Environments](https://mlsanity.com/paper/2109.11990/) (2021)
- [Deconfounded Score Method: Scoring DAGs with Dense Unobserved Confounding](https://mlsanity.com/paper/2103.15106/) (2021)
- [Causality Pursuit from Heterogeneous Environments via Neural Adversarial Invariance Learning](https://mlsanity.com/paper/2405.04715/) (2024)
- [Variance Minimization in the Wasserstein Space for Invariant Causal Prediction](https://mlsanity.com/paper/2110.07064/) (2021)
- [Causality-oriented robustness: exploiting general noise interventions](https://mlsanity.com/paper/2307.10299/) (2023)
- [PatchMix Augmentation to Identify Causal Features in Few-shot Learning](https://mlsanity.com/paper/2211.16019/) (2022)
- [PORCA: Root Cause Analysis with Partially Observed Data](https://mlsanity.com/paper/2407.05869/) (2024)
- [Sample Selection Bias in Evaluation of Prediction Performance of Causal Models](https://mlsanity.com/paper/2106.01921/) (2021)
- [Invariant Probabilistic Prediction](https://mlsanity.com/paper/2309.10083/) (2023)

## Abstract (arXiv)

Causal inference is known to be very challenging when only observational data are available. Randomized experiments are often costly and impractical and in instrumental variable regression the number of instruments has to exceed the number of causal predictors. It was recently shown in Peters et al. [2016] that causal inference for the full model is possible when data from distinct observational environments are available, exploiting that the conditional distribution of a response variable is invariant under the correct causal model. Two shortcomings of such an approach are the high computational effort for large-scale data and the assumed absence of hidden confounders. Here we show that these two shortcomings can be addressed if one is willing to make a more restrictive assumption on the type of interventions that generate different environments. Thereby, we look at a different notion of invariance, namely inner-product invariance. By avoiding a computationally cumbersome reverse-engineering approach such as in Peters et al. [2016], it allows for large-scale causal inference in linear structural equation models. We discuss identifiability conditions for the causal parameter and derive asymptotic confidence intervals in the low-dimensional setting. In the case of non-identifiability we show that the solution set of causal Dantzig has predictive guarantees under certain interventions. We derive finite-sample bounds in the high-dimensional setting and investigate its performance on simulated datasets.