# Recursive Bayesian Networks: Generalising and Unifying Probabilistic Context-Free Grammars and Dynamic Bayesian Networks

arXiv: https://arxiv.org/abs/2111.01853 · HTML: https://mlsanity.com/paper/2111.01853/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 90 references that are in the map.
- First posted on arXiv 2 Nov 2021.

## Builds on (most cited first, 17 of 17)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Graph Structure Learning for Robust Graph Neural Networks](https://mlsanity.com/paper/2005.10203/) (2020)
- [DAG-GNN: DAG Structure Learning with Graph Neural Networks](https://mlsanity.com/paper/1904.10098/) (2019)
- [Learning Discrete Structures for Graph Neural Networks](https://mlsanity.com/paper/1903.11960/) (2019)
- [Learning Bayesian Networks: The Combination of Knowledge and Statistical Data](https://mlsanity.com/paper/1302.6815/) (2013)
- [Compound Probabilistic Context-Free Grammars for Grammar Induction](https://mlsanity.com/paper/1906.10225/) (2019)
- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [Dirichlet-based Gaussian Processes for Large-scale Calibrated Classification](https://mlsanity.com/paper/1805.10915/) (2018)
- [Fast and Accurate Neural CRF Constituency Parsing](https://mlsanity.com/paper/2008.03736/) (2020)
- [Torch-Struct: Deep Structured Prediction Library](https://mlsanity.com/paper/2002.00876/) (2020)
- [Conditional Sum-Product Networks: Imposing Structure on Deep Probabilistic Architectures](https://mlsanity.com/paper/1905.08550/) (2019)
- [Bayesian Learning of Sum-Product Networks](https://mlsanity.com/paper/1905.10884/) (2019)
- [Learning Bayesian Networks: A Unification for Discrete and Gaussian Domains](https://mlsanity.com/paper/1302.4957/) (2013)
- [Automatic Bayesian Density Analysis](https://mlsanity.com/paper/1807.09306/) (2018)
- [Guaranteed Scalable Learning of Latent Tree Models](https://mlsanity.com/paper/1406.4566/) (2014)
- [Factor Graph Grammars](https://mlsanity.com/paper/2010.12048/) (2020)
- [Gaussian Mixture Latent Vector Grammars](https://mlsanity.com/paper/1805.04688/) (2018)

## Built on by (most cited first, 3 of 3)

- [GFlowNet-EM for learning compositional latent variable models](https://mlsanity.com/paper/2302.06576/) (2023)
- [Rethinking the Relationship between the Power Law and Hierarchical Structures](https://mlsanity.com/paper/2505.04984/) (2025)
- [Bayesian Decision Trees via Tractable Priors and Probabilistic Context-Free Grammars](https://mlsanity.com/paper/2302.07407/) (2023)

## Abstract (arXiv)

Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. 2) For Gaussian RBNs, we additionally derive an analytic approximation, allowing for robust parameter optimisation and Bayesian inference. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations.