# Visualizing and Understanding Sum-Product Networks

arXiv: https://arxiv.org/abs/1608.08266 · HTML: https://mlsanity.com/paper/1608.08266/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Aug 2016.

## Built on by (most cited first, 18 of 18)

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- [Sum-Product-Set Networks: Deep Tractable Models for Tree-Structured Graphs](https://mlsanity.com/paper/2408.07394/) (2024)
- [GraphSPNs: Sum-Product Networks Benefit From Canonical Orderings](https://mlsanity.com/paper/2408.09451/) (2024)
- [On Faster Marginalization with Squared Circuits via Orthonormalization](https://mlsanity.com/paper/2412.07883/) (2024)
- [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)
- [Probabilistic Graph Circuits: Deep Generative Models for Tractable Probabilistic Inference over Graphs](https://mlsanity.com/paper/2503.12162/) (2025)
- [Counterfactual Explanations in Medical Imaging: Exploring SPN-Guided Latent Space Manipulation](https://mlsanity.com/paper/2507.19368/) (2025)

## Abstract (arXiv)

Sum-Product Networks (SPNs) are recently introduced deep tractable probabilistic models by which several kinds of inference queries can be answered exactly and in a tractable time. Up to now, they have been largely used as black box density estimators, assessed only by comparing their likelihood scores only. In this paper we explore and exploit the inner representations learned by SPNs. We do this with a threefold aim: first we want to get a better understanding of the inner workings of SPNs; secondly, we seek additional ways to evaluate one SPN model and compare it against other probabilistic models, providing diagnostic tools to practitioners; lastly, we want to empirically evaluate how good and meaningful the extracted representations are, as in a classic Representation Learning framework. In order to do so we revise their interpretation as deep neural networks and we propose to exploit several visualization techniques on their node activations and network outputs under different types of inference queries. To investigate these models as feature extractors, we plug some SPNs, learned in a greedy unsupervised fashion on image datasets, in supervised classification learning tasks. We extract several embedding types from node activations by filtering nodes by their type, by their associated feature abstraction level and by their scope. In a thorough empirical comparison we prove them to be competitive against those generated from popular feature extractors as Restricted Boltzmann Machines. Finally, we investigate embeddings generated from random probabilistic marginal queries as means to compare other tractable probabilistic models on a common ground, extending our experiments to Mixtures of Trees.