# Empirical Bayesian Approaches for Robust Constraint-based Causal Discovery under Insufficient Data

arXiv: https://arxiv.org/abs/2206.08448 · HTML: https://mlsanity.com/paper/2206.08448/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 47 references that are in the map.
- First posted on arXiv 16 Jun 2022.

## Builds on (most cited first, 6 of 6)

- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Approximate Kernel-based Conditional Independence Tests for Fast Non-Parametric Causal Discovery](https://mlsanity.com/paper/1702.03877/) (2017)
- [Model-Powered Conditional Independence Test](https://mlsanity.com/paper/1709.06138/) (2017)
- [Bayesian Structure Learning by Recursive Bootstrap](https://mlsanity.com/paper/1809.04828/) (2018)
- [Towards Efficient Local Causal Structure Learning](https://mlsanity.com/paper/2103.00378/) (2021)
- [A Single Iterative Step for Anytime Causal Discovery](https://mlsanity.com/paper/2012.07513/) (2020)

## Built on by (most cited first, 2 of 2)

- [Bayesian Network Structure Discovery Using Large Language Models](https://mlsanity.com/paper/2511.00574/) (2025)
- [Consistency evaluation of benchmarks used for causal discovery](https://mlsanity.com/paper/2606.01789/) (2026)

## Abstract (arXiv)

Causal discovery is to learn cause-effect relationships among variables given observational data and is important for many applications. Existing causal discovery methods assume data sufficiency, which may not be the case in many real world datasets. As a result, many existing causal discovery methods can fail under limited data. In this work, we propose Bayesian-augmented frequentist independence tests to improve the performance of constraint-based causal discovery methods under insufficient data: 1) We firstly introduce a Bayesian method to estimate mutual information (MI), based on which we propose a robust MI based independence test; 2) Secondly, we consider the Bayesian estimation of hypothesis likelihood and incorporate it into a well-defined statistical test, resulting in a robust statistical testing based independence test. We apply proposed independence tests to constraint-based causal discovery methods and evaluate the performance on benchmark datasets with insufficient samples. Experiments show significant performance improvement in terms of both accuracy and efficiency over SOTA methods.