# Causal Inference Despite Limited Global Confounding via Mixture Models

arXiv: https://arxiv.org/abs/2112.11602 · HTML: https://mlsanity.com/paper/2112.11602/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 22 Dec 2021.

## Builds on (most cited first, 10 of 10)

- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)
- [Learning latent causal graphs via mixture oracles](https://mlsanity.com/paper/2106.15563/) (2021)
- [Comment on "Blessings of Multiple Causes"](https://mlsanity.com/paper/1910.05438/) (2019)
- [Beyond the Low-Degree Algorithm: Mixtures of Subcubes and Their Applications](https://mlsanity.com/paper/1803.06521/) (2018)
- [Multiple Causal Inference with Latent Confounding](https://mlsanity.com/paper/1805.08273/) (2018)
- [Source Identification for Mixtures of Product Distributions](https://mlsanity.com/paper/2012.14540/) (2020)
- [Learning Mixtures of DAG Models](https://mlsanity.com/paper/1301.7415/) (2013)
- [Learning Overcomplete HMMs](https://mlsanity.com/paper/1711.02309/) (2017)
- [Hadamard Extensions and the Identification of Mixtures of Product Distributions](https://mlsanity.com/paper/2101.11688/) (2021)

## Built on by (most cited first, 9 of 9)

- [Identification of Mixtures of Discrete Product Distributions in Near-Optimal Sample and Time Complexity](https://mlsanity.com/paper/2309.13993/) (2023)
- [Causal Discovery in Mixtures of Populations](https://mlsanity.com/paper/2311.07454/) (2023)
- [Identifiability of Product of Experts Models](https://mlsanity.com/paper/2310.09397/) (2023)
- [Synthetic Potential Outcomes and Causal Mixture Identifiability](https://mlsanity.com/paper/2405.19225/) (2024)
- [Learning Discrete Latent Variable Structures with Tensor Rank Conditions](https://mlsanity.com/paper/2406.07020/) (2024)
- [Controlling for discrete unmeasured confounding in nonlinear causal models](https://mlsanity.com/paper/2408.05647/) (2024)
- [On Theoretical Identifiability of Discrete Latent Causal Graphical Models](https://mlsanity.com/paper/2505.18410/) (2025)
- [Mixture Proportion Estimation and Weakly-supervised Kernel Test for Conditional Independence](https://mlsanity.com/paper/2604.07191/) (2026)
- [Causal Inference with Unobserved Confounding: A Mixture Learning Perspective](https://mlsanity.com/paper/2609.23219/) (2026)

## Abstract (arXiv)

A Bayesian Network is a directed acyclic graph (DAG) on a set of n random variables (the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the random variables that is Markovian on the graph. A finite k-mixture of such models is graphically represented by a larger graph which has an additional ``hidden'' (or ``latent'') random variable U, ranging in 1,ldots,k, and a directed edge from U to every other vertex. Models of this type are fundamental to causal inference, where U models an unobserved confounding effect of multiple populations, obscuring the causal relationships in the observable DAG. By solving the mixture problem and recovering the joint probability distribution with U, traditionally unidentifiable causal relationships become identifiable. Using a reduction to the more well-studied ``product'' case on empty graphs, we give the first algorithm to learn mixtures of non-empty DAGs.