# Continuous Bayesian Model Selection for Multivariate Causal Discovery

arXiv: https://arxiv.org/abs/2411.10154 · HTML: https://mlsanity.com/paper/2411.10154/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 72 references that are in the map.
- First posted on arXiv 15 Nov 2024.

## Builds on (most cited first, 30 of 34)

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [CAM: Causal additive models, high-dimensional order search and penalized regression](https://mlsanity.com/paper/1310.1533/) (2013)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Learning Neural Causal Models from Unknown Interventions](https://mlsanity.com/paper/1910.01075/) (2019)
- [Score matching enables causal discovery of nonlinear additive noise models](https://mlsanity.com/paper/2203.04413/) (2022)
- [DiBS: Differentiable Bayesian Structure Learning](https://mlsanity.com/paper/2105.11839/) (2021)
- [Efficient Neural Causal Discovery without Acyclicity Constraints](https://mlsanity.com/paper/2107.10483/) (2021)
- [Causal machine learning for predicting treatment outcomes](https://mlsanity.com/paper/2410.08770/) (2024)
- [Understanding Probabilistic Sparse Gaussian Process Approximations](https://mlsanity.com/paper/1606.04820/) (2016)
- [BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery](https://mlsanity.com/paper/2112.02761/) (2021)
- [Learning to Induce Causal Structure](https://mlsanity.com/paper/2204.04875/) (2022)
- [DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks](https://mlsanity.com/paper/2010.09133/) (2020)
- [On the Identifiability and Estimation of Causal Location-Scale Noise Models](https://mlsanity.com/paper/2210.09054/) (2022)
- [Convergence of Sparse Variational Inference in Gaussian Processes Regression](https://mlsanity.com/paper/2008.00323/) (2020)
- [DAGs with No Curl: An Efficient DAG Structure Learning Approach](https://mlsanity.com/paper/2106.07197/) (2021)
- [Large-Scale Differentiable Causal Discovery of Factor Graphs](https://mlsanity.com/paper/2206.07824/) (2022)
- [Causal Discovery with Score Matching on Additive Models with Arbitrary Noise](https://mlsanity.com/paper/2304.03265/) (2023)
- [Parameter Priors for Directed Acyclic Graphical Models and the Characterization of Several Probability Distributions](https://mlsanity.com/paper/1301.6697/) (2013)
- [Scaling structural learning with NO-BEARS to infer causal transcriptome networks](https://mlsanity.com/paper/1911.00081/) (2019)
- [Natural Gradients in Practice: Non-Conjugate Variational Inference in Gaussian Process Models](https://mlsanity.com/paper/1803.09151/) (2018)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Stable Differentiable Causal Discovery](https://mlsanity.com/paper/2311.10263/) (2023)
- [Gaussian Process Conditional Density Estimation](https://mlsanity.com/paper/1810.12750/) (2018)

## Built on by (most cited first, 6 of 6)

- [Estimating Interventional Distributions with Uncertain Causal Graphs through Meta-Learning](https://mlsanity.com/paper/2507.05526/) (2025)
- [Use What You Know: Causal Foundation Models with Partial Graphs](https://mlsanity.com/paper/2602.14972/) (2026)
- [Identifying Causal Direction via Variational Bayesian Compression](https://mlsanity.com/paper/2505.07503/) (2025)
- [Consistency evaluation of benchmarks used for causal discovery](https://mlsanity.com/paper/2606.01789/) (2026)
- [PRIM: Meta-Learned Bayesian Root Cause Analysis](https://mlsanity.com/paper/2605.08786/) (2026)
- [Amortized Structured Stochastic Variational Inference for Gaussian Process Latent Variable Models](https://mlsanity.com/paper/2610.03647/) (2026)

## Abstract (arXiv)

Current causal discovery approaches require restrictive model assumptions in the absence of interventional data to ensure structure identifiability. These assumptions often do not hold in real-world applications leading to a loss of guarantees and poor performance in practice. Recent work has shown that, in the bivariate case, Bayesian model selection can greatly improve performance by exchanging restrictive modelling for more flexible assumptions, at the cost of a small probability of making an error. Our work shows that this approach is useful in the important multivariate case as well. We propose a scalable algorithm leveraging a continuous relaxation of the discrete model selection problem. Specifically, we employ the Causal Gaussian Process Conditional Density Estimator (CGP-CDE) as a Bayesian non-parametric model, using its hyperparameters to construct an adjacency matrix. This matrix is then optimised using the marginal likelihood and an acyclicity regulariser, giving the maximum a posteriori causal graph. We demonstrate the competitiveness of our approach, showing it is advantageous to perform multivariate causal discovery without infeasible assumptions using Bayesian model selection.