# Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models

arXiv: https://arxiv.org/abs/2102.10440 · HTML: https://mlsanity.com/paper/2102.10440/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 59 references that are in the map.
- First posted on arXiv 20 Feb 2021.

## Builds on (most cited first, 24 of 24)

- [Counterfactual Fairness](https://mlsanity.com/paper/1703.06856/) (2017)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Counterfactual Off-Policy Evaluation with Gumbel-Max Structural Causal Models](https://mlsanity.com/paper/1905.05824/) (2019)
- [Learning Neural Causal Models from Unknown Interventions](https://mlsanity.com/paper/1910.01075/) (2019)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [DoWhy: An End-to-End Library for Causal Inference](https://mlsanity.com/paper/2011.04216/) (2020)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [Time Series Deconfounder: Estimating Treatment Effects over Time in the Presence of Hidden Confounders](https://mlsanity.com/paper/1902.00450/) (2019)
- [CausalML: Python Package for Causal Machine Learning](https://mlsanity.com/paper/2002.11631/) (2020)
- [Learning Causal Graphs with Small Interventions](https://mlsanity.com/paper/1511.00041/) (2015)
- [Morpho-MNIST: Quantitative Assessment and Diagnostics for Representation Learning](https://mlsanity.com/paper/1809.10780/) (2018)
- [Causal Reasoning from Meta-reinforcement Learning](https://mlsanity.com/paper/1901.08162/) (2019)
- [Learning and Testing Causal Models with Interventions](https://mlsanity.com/paper/1805.09697/) (2018)
- [Conditional Sum-Product Networks: Imposing Structure on Deep Probabilistic Architectures](https://mlsanity.com/paper/1905.08550/) (2019)
- [Bayesian Learning of Sum-Product Networks](https://mlsanity.com/paper/1905.10884/) (2019)
- [Sum-product networks: A survey](https://mlsanity.com/paper/2004.01167/) (2020)
- [On the Relationship between Sum-Product Networks and Bayesian Networks](https://mlsanity.com/paper/1501.01239/) (2015)
- [On the Relative Expressiveness of Bayesian and Neural Networks](https://mlsanity.com/paper/1812.08957/) (2018)
- [Sum-Product-Quotient Networks](https://mlsanity.com/paper/1710.04404/) (2017)
- [Interventions and Counterfactuals in Tractable Probabilistic Models: Limitations of Contemporary Transformations](https://mlsanity.com/paper/2001.10905/) (2020)

## Built on by (most cited first, 22 of 22)

- [Causal Parrots: Large Language Models May Talk Causality But Are Not Causal](https://mlsanity.com/paper/2308.13067/) (2023)
- [Can Foundation Models Talk Causality?](https://mlsanity.com/paper/2206.10591/) (2022)
- [Modeling Causal Mechanisms with Diffusion Models for Interventional and Counterfactual Queries](https://mlsanity.com/paper/2302.00860/) (2023)
- [HyperSPNs: Compact and Expressive Probabilistic Circuits](https://mlsanity.com/paper/2112.00914/) (2021)
- [Compositional Probabilistic and Causal Inference using Tractable Circuit Models](https://mlsanity.com/paper/2304.08278/) (2023)
- [A Compositional Atlas for Algebraic Circuits](https://mlsanity.com/paper/2412.05481/) (2024)
- [Scaling Probabilistic Circuits via Monarch Matrices](https://mlsanity.com/paper/2506.12383/) (2025)
- [Characteristic Circuits](https://mlsanity.com/paper/2312.07790/) (2023)
- [Restructuring Tractable Probabilistic Circuits](https://mlsanity.com/paper/2411.12256/) (2024)
- [Conditional Generative Models are Sufficient to Sample from Any Causal Effect Estimand](https://mlsanity.com/paper/2402.07419/) (2024)
- [CausalMan: A physics-based simulator for large-scale causality](https://mlsanity.com/paper/2502.12707/) (2025)
- [Pearl Causal Hierarchy on Image Data: Intricacies & Challenges](https://mlsanity.com/paper/2212.12570/) (2022)
- [chiSPN: Characteristic Interventional Sum-Product Networks for Causal Inference in Hybrid Domains](https://mlsanity.com/paper/2408.07545/) (2024)
- [GT-CausIn: a novel causal-based insight for traffic prediction](https://mlsanity.com/paper/2212.05782/) (2022)
- [Context-specific Credibility-aware Multimodal Fusion with Conditional Probabilistic Circuits](https://mlsanity.com/paper/2603.26629/) (2026)
- [Explaining Deep Tractable Probabilistic Models: The sum-product network case](https://mlsanity.com/paper/2110.09778/) (2021)
- [A Taxonomy for Inference in Causal Model Families](https://mlsanity.com/paper/2110.12052/) (2021)
- [The Causal Loss: Driving Correlation to Imply Causation](https://mlsanity.com/paper/2110.12066/) (2021)
- [Geometry-Aware Probabilistic Circuits via Voronoi Tessellations](https://mlsanity.com/paper/2603.11946/) (2026)
- [Continual Causal Abstractions](https://mlsanity.com/paper/2212.12575/) (2022)
- [MCNS: Mining Causal Natural Structures Inside Time Series via A Novel Internal Causality Scheme](https://mlsanity.com/paper/2309.06739/) (2023)
- [Interventional Causal Circuits for Safe Robot Action Testing and Failure Recovery](https://mlsanity.com/paper/2607.14826/) (2026)

## Abstract (arXiv)

While probabilistic models are an important tool for studying causality, doing so suffers from the intractability of inference. As a step towards tractable causal models, we consider the problem of learning interventional distributions using sum-product networks (SPNs) that are over-parameterized by gate functions, e.g., neural networks. Providing an arbitrarily intervened causal graph as input, effectively subsuming Pearl's do-operator, the gate function predicts the parameters of the SPN. The resulting interventional SPNs are motivated and illustrated by a structural causal model themed around personal health. Our empirical evaluation on three benchmark data sets as well as a synthetic health data set clearly demonstrates that interventional SPNs indeed are both expressive in modelling and flexible in adapting to the interventions.