# Learning Sparse Fixed-Structure Gaussian Bayesian Networks

arXiv: https://arxiv.org/abs/2107.10450 · HTML: https://mlsanity.com/paper/2107.10450/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 22 Jul 2021.

## Builds on (most cited first, 13 of 13)

- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [On Causal Discovery with Equal Variance Assumption](https://mlsanity.com/paper/1807.03419/) (2018)
- [A polynomial-time algorithm for learning nonparametric causal graphs](https://mlsanity.com/paper/2006.11970/) (2020)
- [Concave Penalized Estimation of Sparse Gaussian Bayesian Networks](https://mlsanity.com/paper/1401.0852/) (2014)
- [Learning Identifiable Gaussian Bayesian Networks in Polynomial Time and Sample Complexity](https://mlsanity.com/paper/1703.01196/) (2017)
- [Testing Bayesian Networks](https://mlsanity.com/paper/1612.03156/) (2016)
- [Multi-Item Mechanisms without Item-Independence: Learnability via Robustness](https://mlsanity.com/paper/1911.02146/) (2019)
- [Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu](https://mlsanity.com/paper/2011.04144/) (2020)
- [Efficient Distance Approximation for Structured High-Dimensional Distributions via Learning](https://mlsanity.com/paper/2002.05378/) (2020)
- [Learning big Gaussian Bayesian networks: partition, estimation, and fusion](https://mlsanity.com/paper/1904.10900/) (2019)
- [Learning Large-Scale Bayesian Networks with the sparsebn Package](https://mlsanity.com/paper/1703.04025/) (2017)
- [Multiple Quantitative Trait Analysis Using Bayesian Networks](https://mlsanity.com/paper/1402.2905/) (2014)
- [Identifiability of Gaussian Structural Equation Models with Homogeneous and Heterogeneous Error Variances](https://mlsanity.com/paper/1901.10134/) (2019)

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

- [Distribution Learning Meets Graph Structure Sampling](https://mlsanity.com/paper/2405.07914/) (2024)
- [Efficient Sample-optimal Learning of Gaussian Tree Models via Sample-optimal Testing of Gaussian Mutual Information](https://mlsanity.com/paper/2411.11516/) (2024)

## Abstract (arXiv)

Gaussian Bayesian networks (a.k.a. linear Gaussian structural equation models) are widely used to model causal interactions among continuous variables. In this work, we study the problem of learning a fixed-structure Gaussian Bayesian network up to a bounded error in total variation distance. We analyze the commonly used node-wise least squares regression (LeastSquares) and prove that it has a near-optimal sample complexity. We also study a couple of new algorithms for the problem: - BatchAvgLeastSquares takes the average of several batches of least squares solutions at each node, so that one can interpolate between the batch size and the number of batches. We show that BatchAvgLeastSquares also has near-optimal sample complexity. - CauchyEst takes the median of solutions to several batches of linear systems at each node. We show that the algorithm specialized to polytrees, CauchyEstTree, has near-optimal sample complexity. Experimentally, we show that for uncontaminated, realizable data, the LeastSquares algorithm performs best, but in the presence of contamination or DAG misspecification, CauchyEst/CauchyEstTree and BatchAvgLeastSquares respectively perform better.