# On the Relationship between Sum-Product Networks and Bayesian Networks

arXiv: https://arxiv.org/abs/1501.01239 · HTML: https://mlsanity.com/paper/1501.01239/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jan 2015.

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

- [Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models](https://mlsanity.com/paper/2102.10440/) (2021)
- [Compositional Probabilistic and Causal Inference using Tractable Circuit Models](https://mlsanity.com/paper/2304.08278/) (2023)
- [Restructuring Tractable Probabilistic Circuits](https://mlsanity.com/paper/2411.12256/) (2024)
- [Neural Probabilistic Circuits: Enabling Compositional and Interpretable Predictions through Logical Reasoning](https://mlsanity.com/paper/2501.07021/) (2025)
- [chiSPN: Characteristic Interventional Sum-Product Networks for Causal Inference in Hybrid Domains](https://mlsanity.com/paper/2408.07545/) (2024)
- [On the Expressive Power of Tree-Structured Probabilistic Circuits](https://mlsanity.com/paper/2410.05465/) (2024)
- [Sum-Product-Attention Networks: Leveraging Self-Attention in Probabilistic Circuits](https://mlsanity.com/paper/2109.06587/) (2021)
- [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)
- [Principled Diverse Counterfactuals in Multilinear Models](https://mlsanity.com/paper/2201.06467/) (2022)
- [Using Sum-Product Networks to Assess Uncertainty in Deep Active Learning](https://mlsanity.com/paper/2206.09798/) (2022)
- [Uncertain Bayesian Networks: Learning from Incomplete Data](https://mlsanity.com/paper/2208.04221/) (2022)
- [A Probabilistic Circuit-Induced Pseudo-Metric for Out-of-Distribution Detection](https://mlsanity.com/paper/2608.09117/) (2026)

## Abstract (arXiv)

In this paper, we establish some theoretical connections between Sum-Product Networks (SPNs) and Bayesian Networks (BNs). We prove that every SPN can be converted into a BN in linear time and space in terms of the network size. The key insight is to use Algebraic Decision Diagrams (ADDs) to compactly represent the local conditional probability distributions at each node in the resulting BN by exploiting context-specific independence (CSI). The generated BN has a simple directed bipartite graphical structure. We show that by applying the Variable Elimination algorithm (VE) to the generated BN with ADD representations, we can recover the original SPN where the SPN can be viewed as a history record or caching of the VE inference process. To help state the proof clearly, we introduce the notion of em normal SPN and present a theoretical analysis of the consistency and decomposability properties. We conclude the paper with some discussion of the implications of the proof and establish a connection between the depth of an SPN and a lower bound of the tree-width of its corresponding BN.