# Structured Neural Networks for Density Estimation and Causal Inference

arXiv: https://arxiv.org/abs/2311.02221 · HTML: https://mlsanity.com/paper/2311.02221/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 52 references that are in the map.
- First posted on arXiv 3 Nov 2023.

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

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Causal-learn: Causal Discovery in Python](https://mlsanity.com/paper/2307.16405/) (2023)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [BCD Nets: Scalable Variational Approaches for Bayesian Causal Discovery](https://mlsanity.com/paper/2112.02761/) (2021)
- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [Graphical Normalizing Flows](https://mlsanity.com/paper/2006.02548/) (2020)
- [Exploration in Structured Reinforcement Learning](https://mlsanity.com/paper/1806.00775/) (2018)
- [Vecchia approximations of Gaussian-process predictions](https://mlsanity.com/paper/1805.03309/) (2018)
- [Normalizing Flows for Interventional Density Estimation](https://mlsanity.com/paper/2209.06203/) (2022)
- [CausalAF: Causal Autoregressive Flow for Safety-Critical Driving Scenario Generation](https://mlsanity.com/paper/2110.13939/) (2021)
- [Partial Identification of Treatment Effects with Implicit Generative Models](https://mlsanity.com/paper/2210.08139/) (2022)
- [An introduction to sampling via measure transport](https://mlsanity.com/paper/1602.05023/) (2016)
- [Personalized Public Policy Analysis in Social Sciences using Causal-Graphical Normalizing Flows](https://mlsanity.com/paper/2202.03281/) (2022)
- [Recent Developments in Boolean Matrix Factorization](https://mlsanity.com/paper/2012.03127/) (2020)
- [Harmonization with Flow-based Causal Inference](https://mlsanity.com/paper/2106.06845/) (2021)
- [Boolean Matrix Factorization and Noisy Completion via Message Passing](https://mlsanity.com/paper/1509.08535/) (2015)
- [Neural Pharmacodynamic State Space Modeling](https://mlsanity.com/paper/2102.11218/) (2021)
- [Copula-Based Deep Survival Models for Dependent Censoring](https://mlsanity.com/paper/2306.11912/) (2023)
- [Parameterized Low-Rank Binary Matrix Approximation](https://mlsanity.com/paper/1803.06102/) (2018)

## Built on by (most cited first, 13 of 13)

- [Hybrid2 Neural ODE Causal Modeling and an Application to Glycemic Response](https://mlsanity.com/paper/2402.17233/) (2024)
- [Causality for Tabular Data Synthesis: A High-Order Structure Causal Benchmark Framework](https://mlsanity.com/paper/2406.08311/) (2024)
- [DeCaFlow: A deconfounding causal generative model](https://mlsanity.com/paper/2503.15114/) (2025)
- [Dimension-independent rates for structured neural density estimation](https://mlsanity.com/paper/2411.15095/) (2024)
- [Breaking the curse of dimensionality in structured density estimation](https://mlsanity.com/paper/2410.07685/) (2024)
- [Towards Reasonable Concept Bottleneck Models](https://mlsanity.com/paper/2506.05014/) (2025)
- [Mitigating Privacy Risk via Forget Set-Free Unlearning](https://mlsanity.com/paper/2604.10636/) (2026)
- [Predicting Long-Term Allograft Survival in Liver Transplant Recipients](https://mlsanity.com/paper/2408.05437/) (2024)
- [Causal Posterior Estimation](https://mlsanity.com/paper/2505.21468/) (2025)
- [Understanding Learning Dynamics Through Structured Representations](https://mlsanity.com/paper/2508.02126/) (2025)
- [CaTs and DAGs: Integrating Directed Acyclic Graphs with Transformers for Causally Constrained Predictions](https://mlsanity.com/paper/2410.14485/) (2024)
- [Neural Autoregressive Flows for Markov Boundary Learning](https://mlsanity.com/paper/2603.20791/) (2026)
- [Steering Neural Network Training through Interpretable Constraints Based on Partial Dependence](https://mlsanity.com/paper/2607.08641/) (2026)

## Abstract (arXiv)

Injecting structure into neural networks enables learning functions that satisfy invariances with respect to subsets of inputs. For instance, when learning generative models using neural networks, it is advantageous to encode the conditional independence structure of observed variables, often in the form of Bayesian networks. We propose the Structured Neural Network (StrNN), which injects structure through masking pathways in a neural network. The masks are designed via a novel relationship we explore between neural network architectures and binary matrix factorization, to ensure that the desired independencies are respected. We devise and study practical algorithms for this otherwise NP-hard design problem based on novel objectives that control the model architecture. We demonstrate the utility of StrNN in three applications: (1) binary and Gaussian density estimation with StrNN, (2) real-valued density estimation with Structured Autoregressive Flows (StrAFs) and Structured Continuous Normalizing Flows (StrCNF), and (3) interventional and counterfactual analysis with StrAFs for causal inference. Our work opens up new avenues for learning neural networks that enable data-efficient generative modeling and the use of normalizing flows for causal effect estimation.