mlsanityData to 14 Oct 2026

BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery

arXiv 2112.02761 ↗First posted 6 Dec 2021cs.LG cs.AI stat.MLcited by 68

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Abstract (arXiv)

A structural equation model (SEM) is an effective framework to reason over causal relationships represented via a directed acyclic graph (DAG). Recent advances have enabled effective maximum-likelihood point estimation of DAGs from observational data. However, a point estimate may not accurately capture the uncertainty in inferring the underlying graph in practical scenarios, wherein the true DAG is non-identifiable and/or the observed dataset is limited. We propose Bayesian Causal Discovery Nets (BCD Nets), a variational inference framework for estimating a distribution over DAGs characterizing a linear-Gaussian SEM. Developing a full Bayesian posterior over DAGs is challenging due to the the discrete and combinatorial nature of graphs. We analyse key design choices for scalable VI over DAGs, such as 1) the parametrization of DAGs via an expressive variational family, 2) a continuous relaxation that enables low-variance stochastic optimization, and 3) suitable priors over the latent variables. We provide a series of experiments on real and synthetic data showing that BCD Nets outperform maximum-likelihood methods on standard causal discovery metrics such as structural Hamming distance in low data regimes.

Key facts. Cited by 68 papers in the map (data through 14 Oct 2026). Builds on 22 of its 73 references that are in the map. First posted on arXiv 6 Dec 2021.

Builds on

22 of 73 references.

References · 22

  1. 2015Variational Inference with Normalizing Flowscited by 2,170
  2. 2013Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computationcited by 2,079
  3. 2016Variational Inference: A Review for Statisticianscited by 1,472
  4. 2018DAGs with NO TEARS: Continuous Optimization for Structure Learningcited by 752
  5. 2019DAG-GNN: DAG Structure Learning with Graph Neural Networkscited by 331
  6. 2015MADE: Masked Autoencoder for Distribution Estimationcited by 307
  7. 2019Monte Carlo Gradient Estimation in Machine Learningcited by 283
  8. 2020AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradientscited by 283
  9. 2013Causal Discovery with Continuous Additive Noise Modelscited by 255
  10. 2020On the Role of Sparsity and DAG Constraints for Learning Linear DAGscited by 191
  11. 2018Learning Latent Permutations with Gumbel-Sinkhorn Networkscited by 175
  12. 2017Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inferencecited by 98
10 more
  1. 2020DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networkscited by 66
  2. 2017Learning linear structural equation models in polynomial time and sample complexitycited by 51
  3. 2020Towards Scalable Bayesian Learning of Causal DAGscited by 33
  4. 2015Bayesian model averaging: A systematic review and conceptual classificationcited by 29
  5. 2019Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytopecited by 26
  6. 2020Locally Masked Convolution for Autoregressive Modelscited by 22
  7. 2016On the Hyperprior Choice for the Global Shrinkage Parameter in the Horseshoe Priorcited by 10
  8. 2016Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets, and Complexitycited by 9
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 15; 2023: 21; 2024: 15; 2025: 5; 2026: 12.

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  1. 2022Bayesian Structure Learning with Generative Flow Networkscited by 153
  2. 2023A theory of continuous generative flow networkscited by 101
  3. 2022Deep End-to-end Causal Inferencecited by 95
  4. 2023GFlowNets for AI-Driven Scientific Discoverycited by 63
  5. 2023Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Networkcited by 50
  6. 2022Interventions, Where and How? Experimental Design for Causal Models at Scalecited by 49
  7. 2023CUTS+: High-dimensional Causal Discovery from Irregular Time-seriescited by 41
  8. 2022Active Bayesian Causal Inferencecited by 35
  9. 2023BayesDAG: Gradient-Based Posterior Inference for Causal Discoverycited by 33
  10. 2022Bayesian learning of Causal Structure and Mechanisms with GFlowNets and Variational Bayescited by 30
  11. 2023DynGFN: Towards Bayesian Inference of Gene Regulatory Networks with GFlowNetscited by 27
  12. 2022BaCaDI: Bayesian Causal Discovery with Unknown Interventionscited by 26
38 more
  1. 2024A Meta-Learning Approach to Bayesian Causal Discoverycited by 22
  2. 2022Tractable Uncertainty for Structure Learningcited by 15
  3. 2023Differentiable Multi-Target Causal Bayesian Experimental Designcited by 15
  4. 2023Finding Counterfactually Optimal Action Sequences in Continuous State Spacescited by 13
  5. 2023CoLiDE: Concomitant Linear DAG Estimationcited by 13
  6. 2023Structured Neural Networks for Density Estimation and Causal Inferencecited by 13
  7. 2023Constraint-Free Structure Learning with Smooth Acyclic Orientationscited by 10
  8. 2023DAG Learning on the Permutahedroncited by 9
  9. 2022On the Generalization and Adaption Performance of Causal Modelscited by 8
  10. 2024Effective Bayesian Causal Inference via Structural Marginalisation and Autoregressive Orderscited by 6
  11. 2024Scalable Differentiable Causal Discovery in the Presence of Latent Confounders with Skeleton Posterior (Extended Version)cited by 6
  12. 2024Continuous Bayesian Model Selection for Multivariate Causal Discoverycited by 6
  13. 2022Learning Latent Structural Causal Modelscited by 5
  14. 2022Trust Your nabla: Gradient-based Intervention Targeting for Causal Discoverycited by 5
  15. 2023Order-based Structure Learning with Normalizing Flowscited by 5
  16. 2023Causal Inference in Gene Regulatory Networks with GFlowNet: Towards Scalability in Large Systemscited by 5
  17. 2024Challenges and Considerations in the Evaluation of Bayesian Causal Discoverycited by 5
  18. 2023Structure Learning with Adaptive Random Neighborhood Informed MCMCcited by 4
  19. 2024Permutation-based Inference for Variational Learning of Directed Acyclic Graphscited by 4
  20. 2024ProDAG: Projected Variational Inference for Directed Acyclic Graphscited by 4
  21. 2024Sample Efficient Bayesian Learning of Causal Graphs from Interventionscited by 4
  22. 2022Learning Multiscale Non-stationary Causal Structurescited by 3
  23. 2022Learning Discrete Directed Acyclic Graphs via Backpropagationcited by 3
  24. 2023Benchmarking Bayesian Causal Discovery Methods for Downstream Treatment Effect Estimationcited by 3
  25. 2023Differentiable Bayesian Structure Learning with Acyclicity Assurancecited by 3
  26. 2024Reinforcement Learning for Causal Discovery without Acyclicity Constraintscited by 3
  27. 2025Identifying Causal Direction via Variational Bayesian Compressioncited by 3
  28. 2022GT-CausIn: a novel causal-based insight for traffic predictioncited by 2
  29. 2023Causal structure learning with momentum: Sampling distributions over Markov Equivalence Classes of DAGscited by 2
  30. 2024OCDB: Revisiting Causal Discovery with a Comprehensive Benchmark and Evaluation Frameworkcited by 2
  31. 2025Goal-Oriented Sequential Bayesian Experimental Design for Causal Learningcited by 2
  32. 2026Concomitant DAG Learning: On the Roles of Noise Adaptivity, Sparsity, and Non-negativitycited by 2
  33. 2022Latent Variable Models for Bayesian Causal Discoverycited by 1
  34. 2022Learning Large Causal Structures from Inverse Covariance Matrix via Sparse Matrix Decompositioncited by 1
  35. 2023Causal Graph Discovery from Self and Mutually Exciting Time Seriescited by 1
  36. 2023Discovering Dynamic Effective Connectome of Brain with Bayesian Dynamic DAG Learningcited by 1
  37. 2023Towards Human-like Perception: Learning Structural Causal Model in Heterogeneous Graphcited by 1
  38. 2024Enabling Causal Discovery in Post-Nonlinear Models with Normalizing Flowscited by 1

50 most cited of 68 All 68 (JSON)