# Conditional Sum-Product Networks: Imposing Structure on Deep Probabilistic Architectures

arXiv: https://arxiv.org/abs/1905.08550 · HTML: https://mlsanity.com/paper/1905.08550/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2019.

## Built on by (most cited first, 24 of 24)

- [Semantic Probabilistic Layers for Neuro-Symbolic Learning](https://mlsanity.com/paper/2206.00426/) (2022)
- [Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models](https://mlsanity.com/paper/2102.10440/) (2021)
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- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [Continuous Mixtures of Tractable Probabilistic Models](https://mlsanity.com/paper/2209.10584/) (2022)
- [HyperSPNs: Compact and Expressive Probabilistic Circuits](https://mlsanity.com/paper/2112.00914/) (2021)
- [Breaking the Factorization Barrier in Diffusion Language Models](https://mlsanity.com/paper/2603.00045/) (2026)
- [Fast and Expressive Multi-Byte Prediction with Probabilistic Circuits](https://mlsanity.com/paper/2511.11346/) (2025)
- [Leveraging Probabilistic Circuits for Nonparametric Multi-Output Regression](https://mlsanity.com/paper/2106.08687/) (2021)
- [Probabilistic Integral Circuits](https://mlsanity.com/paper/2310.16986/) (2023)
- [Defining neurosymbolic AI](https://mlsanity.com/paper/2507.11127/) (2025)
- [Tractable Probabilistic Graph Representation Learning with Graph-Induced Sum-Product Networks](https://mlsanity.com/paper/2305.10544/) (2023)
- [A Unified Framework for Human-Allied Learning of Probabilistic Circuits](https://mlsanity.com/paper/2405.02413/) (2024)
- [Deconvolutional Density Network: Modeling Free-Form Conditional Distributions](https://mlsanity.com/paper/2105.14367/) (2021)
- [Recursive Bayesian Networks: Generalising and Unifying Probabilistic Context-Free Grammars and Dynamic Bayesian Networks](https://mlsanity.com/paper/2111.01853/) (2021)
- [Efficient and Reliable Probabilistic Interactive Learning with Structured Outputs](https://mlsanity.com/paper/2202.08566/) (2022)
- [Context-specific Credibility-aware Multimodal Fusion with Conditional Probabilistic Circuits](https://mlsanity.com/paper/2603.26629/) (2026)
- [Sum-Product-Attention Networks: Leveraging Self-Attention in Probabilistic Circuits](https://mlsanity.com/paper/2109.06587/) (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)
- [TreeFlow: Going beyond Tree-based Gaussian Probabilistic Regression](https://mlsanity.com/paper/2206.04140/) (2022)
- [RECOWNs: Probabilistic Circuits for Trustworthy Time Series Forecasting](https://mlsanity.com/paper/2106.04148/) (2021)
- [Learning Tractable Distributions Of Language Model Continuations](https://mlsanity.com/paper/2511.16054/) (2025)
- [Reliable Fusion of Conflicting Experts](https://mlsanity.com/paper/2609.27913/) (2026)

## Abstract (arXiv)

Probabilistic graphical models are a central tool in AI; however, they are generally not as expressive as deep neural models, and inference is notoriously hard and slow. In contrast, deep probabilistic models such as sum-product networks (SPNs) capture joint distributions in a tractable fashion, but still lack the expressive power of intractable models based on deep neural networks. Therefore, we introduce conditional SPNs (CSPNs), conditional density estimators for multivariate and potentially hybrid domains which allow harnessing the expressive power of neural networks while still maintaining tractability guarantees. One way to implement CSPNs is to use an existing SPN structure and condition its parameters on the input, e.g., via a deep neural network. This approach, however, might misrepresent the conditional independence structure present in data. Consequently, we also develop a structure-learning approach that derives both the structure and parameters of CSPNs from data. Our experimental evidence demonstrates that CSPNs are competitive with other probabilistic models and yield superior performance on multilabel image classification compared to mean field and mixture density networks. Furthermore, they can successfully be employed as building blocks for structured probabilistic models, such as autoregressive image models.