# On the Relative Expressiveness of Bayesian and Neural Networks

arXiv: https://arxiv.org/abs/1812.08957 · HTML: https://mlsanity.com/paper/1812.08957/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Dec 2018.

## Built on by (most cited first, 7 of 7)

- [Foundations of Symbolic Languages for Model Interpretability](https://mlsanity.com/paper/2110.02376/) (2021)
- [Interventional Sum-Product Networks: Causal Inference with Tractable Probabilistic Models](https://mlsanity.com/paper/2102.10440/) (2021)
- [Tractable Boolean and Arithmetic Circuits](https://mlsanity.com/paper/2202.02942/) (2022)
- [chiSPN: Characteristic Interventional Sum-Product Networks for Causal Inference in Hybrid Domains](https://mlsanity.com/paper/2408.07545/) (2024)
- [Independence Is Not an Issue in Neurosymbolic AI](https://mlsanity.com/paper/2504.07851/) (2025)
- [Exact Learning Augmented Naive Bayes Classifier](https://mlsanity.com/paper/2107.03018/) (2021)
- [Learning Tractable Distributions Of Language Model Continuations](https://mlsanity.com/paper/2511.16054/) (2025)

## Abstract (arXiv)

A neural network computes a function. A central property of neural networks is that they are "universal approximators:" for a given continuous function, there exists a neural network that can approximate it arbitrarily well, given enough neurons (and some additional assumptions). In contrast, a Bayesian network is a model, but each of its queries can be viewed as computing a function. In this paper, we identify some key distinctions between the functions computed by neural networks and those by marginal Bayesian network queries, showing that the former are more expressive than the latter. Moreover, we propose a simple augmentation to Bayesian networks (a testing operator), which enables their marginal queries to become "universal approximators."