# ProDAG: Projected Variational Inference for Directed Acyclic Graphs

arXiv: https://arxiv.org/abs/2405.15167 · HTML: https://mlsanity.com/paper/2405.15167/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 46 references that are in the map.
- First posted on arXiv 24 May 2024.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [D'ya like DAGs? A Survey on Structure Learning and Causal Discovery](https://mlsanity.com/paper/2103.02582/) (2021)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [DiBS: Differentiable Bayesian Structure Learning](https://mlsanity.com/paper/2105.11839/) (2021)
- [Deep End-to-end Causal Inference](https://mlsanity.com/paper/2202.02195/) (2022)
- [A survey of Bayesian Network structure learning](https://mlsanity.com/paper/2109.11415/) (2021)
- [BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery](https://mlsanity.com/paper/2112.02761/) (2021)
- [Fashionable Modelling with Flux](https://mlsanity.com/paper/1811.01457/) (2018)
- [DAGs with No Curl: An Efficient DAG Structure Learning Approach](https://mlsanity.com/paper/2106.07197/) (2021)
- [Joint Bayesian Inference of Graphical Structure and Parameters with a Single Generative Flow Network](https://mlsanity.com/paper/2305.19366/) (2023)
- [Differentiable DAG Sampling](https://mlsanity.com/paper/2203.08509/) (2022)
- [Large-Scale Differentiable Causal Discovery of Factor Graphs](https://mlsanity.com/paper/2206.07824/) (2022)
- [Variational Causal Networks: Approximate Bayesian Inference over Causal Structures](https://mlsanity.com/paper/2106.07635/) (2021)
- [Fast Scalable and Accurate Discovery of DAGs Using the Best Order Score Search and Grow-Shrink Trees](https://mlsanity.com/paper/2310.17679/) (2023)
- [Towards Scalable Bayesian Learning of Causal DAGs](https://mlsanity.com/paper/2010.00684/) (2020)
- [BayesDAG: Gradient-Based Posterior Inference for Causal Discovery](https://mlsanity.com/paper/2307.13917/) (2023)
- [Differentiable Causal Discovery Under Latent Interventions](https://mlsanity.com/paper/2203.02336/) (2022)
- [Integer Programming for Learning Directed Acyclic Graphs from Continuous Data](https://mlsanity.com/paper/1904.10574/) (2019)
- [Optimizing NOTEARS Objectives via Topological Swaps](https://mlsanity.com/paper/2305.17277/) (2023)
- [Tractable Uncertainty for Structure Learning](https://mlsanity.com/paper/2204.14170/) (2022)
- [Differentiable Multi-Target Causal Bayesian Experimental Design](https://mlsanity.com/paper/2302.10607/) (2023)
- [Effective Bayesian Causal Inference via Structural Marginalisation and Autoregressive Orders](https://mlsanity.com/paper/2402.14781/) (2024)
- [Permutation-based Inference for Variational Learning of Directed Acyclic Graphs](https://mlsanity.com/paper/2402.02644/) (2024)
- [Contextual Directed Acyclic Graphs](https://mlsanity.com/paper/2310.15627/) (2023)
- [Learning Large DAGs by Combining Continuous Optimization and Feedback Arc Set Heuristics](https://mlsanity.com/paper/2107.00571/) (2021)

## Built on by (most cited first, 4 of 4)

- [Arrow: A Foundation Model for Causal Discovery](https://mlsanity.com/paper/2605.07204/) (2026)
- [Causal Atlases from Entropic Inference: Bayesian Networks beyond Optimal DAGs](https://mlsanity.com/paper/2606.06440/) (2026)
- [Structure Learning on Clustered Data](https://mlsanity.com/paper/2607.08238/) (2026)
- [SVI-DAG: A Structured Variational Inference Approach to Bayesian Causal Discovery](https://mlsanity.com/paper/2608.04930/) (2026)

## Abstract (arXiv)

Directed acyclic graph (DAG) learning is a central task in structure discovery and causal inference. Although the field has witnessed remarkable advances over the past few years, it remains statistically and computationally challenging to learn a single (point estimate) DAG from data, let alone provide uncertainty quantification. We address the difficult task of quantifying graph uncertainty by developing a Bayesian variational inference framework based on novel, provably valid distributions that have support directly on the space of sparse DAGs. These distributions, which we use to define our prior and variational posterior, are induced by a projection operation that maps an arbitrary continuous distribution onto the space of sparse weighted acyclic adjacency matrices. While this projection is combinatorial, it can be solved efficiently using recent continuous reformulations of acyclicity constraints. We empirically demonstrate that our method, ProDAG, can outperform state-of-the-art alternatives in both accuracy and uncertainty quantification.