# Compositional Probabilistic and Causal Inference using Tractable Circuit Models

arXiv: https://arxiv.org/abs/2304.08278 · HTML: https://mlsanity.com/paper/2304.08278/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 17 Apr 2023.

## Builds on (most cited first, 14 of 14)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits](https://mlsanity.com/paper/2004.06231/) (2020)
- [A Differential Approach to Inference in Bayesian Networks](https://mlsanity.com/paper/1301.3847/) (2013)
- [On the Tractability of SHAP Explanations](https://mlsanity.com/paper/2009.08634/) (2020)
- [Strudel: Learning Structured-Decomposable Probabilistic Circuits](https://mlsanity.com/paper/2007.09331/) (2020)
- [Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models](https://mlsanity.com/paper/2102.10440/) (2021)
- [On Relaxing Determinism in Arithmetic Circuits](https://mlsanity.com/paper/1708.06846/) (2017)
- [Smoothing Structured Decomposable Circuits](https://mlsanity.com/paper/1906.00311/) (2019)
- [On the Relationship between Sum-Product Networks and Bayesian Networks](https://mlsanity.com/paper/1501.01239/) (2015)
- [Causal Inference Using Tractable Circuits](https://mlsanity.com/paper/2202.02891/) (2022)
- [Solving Marginal MAP Exactly by Probabilistic Circuit Transformations](https://mlsanity.com/paper/2111.04833/) (2021)
- [Provable Guarantees on the Robustness of Decision Rules to Causal Interventions](https://mlsanity.com/paper/2105.09108/) (2021)
- [Sum-Product Network Decompilation](https://mlsanity.com/paper/1912.10092/) (2019)
- [Interventions and Counterfactuals in Tractable Probabilistic Models: Limitations of Contemporary Transformations](https://mlsanity.com/paper/2001.10905/) (2020)

## Built on by (most cited first, 10 of 10)

- [Scaling Tractable Probabilistic Circuits: A Systems Perspective](https://mlsanity.com/paper/2406.00766/) (2024)
- [On the Relationship Between Monotone and Squared Probabilistic Circuits](https://mlsanity.com/paper/2408.00876/) (2024)
- [Building Expressive and Tractable Probabilistic Generative Models: A Review](https://mlsanity.com/paper/2402.00759/) (2024)
- [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)
- [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)
- [Interpretable Neural Causal Models with TRAM-DAGs](https://mlsanity.com/paper/2503.16206/) (2025)
- [A canonical generalization of OBDD](https://mlsanity.com/paper/2604.05537/) (2026)
- [Interventional Causal Circuits for Safe Robot Action Testing and Failure Recovery](https://mlsanity.com/paper/2607.14826/) (2026)

## Abstract (arXiv)

Probabilistic circuits (PCs) are a class of tractable probabilistic models, which admit efficient inference routines depending on their structural properties. In this paper, we introduce md-vtrees, a novel structural formulation of (marginal) determinism in structured decomposable PCs, which generalizes previously proposed classes such as probabilistic sentential decision diagrams. Crucially, we show how mdvtrees can be used to derive tractability conditions and efficient algorithms for advanced inference queries expressed as arbitrary compositions of basic probabilistic operations, such as marginalization, multiplication and reciprocals, in a sound and generalizable manner. In particular, we derive the first polytime algorithms for causal inference queries such as backdoor adjustment on PCs. As a practical instantiation of the framework, we propose MDNets, a novel PC architecture using md-vtrees, and empirically demonstrate their application to causal inference.