# An Empirical-Bayes Score for Discrete Bayesian Networks

arXiv: https://arxiv.org/abs/1605.03884 · HTML: https://mlsanity.com/paper/1605.03884/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 May 2016.

## Built on by (most cited first, 9 of 9)

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- [Causal discovery using dynamically requested knowledge](https://mlsanity.com/paper/2310.11154/) (2023)
- [Hybrid Bayesian network discovery with latent variables by scoring multiple interventions](https://mlsanity.com/paper/2112.10574/) (2021)
- [Parallel Sampling for Efficient High-dimensional Bayesian Network Structure Learning](https://mlsanity.com/paper/2202.09691/) (2022)
- [ReBeCA: Unveiling Interpretable Behavior Hierarchy behind the Iterative Self-Reflection of Language Models with Causal Analysis](https://mlsanity.com/paper/2602.06373/) (2026)

## Abstract (arXiv)

Bayesian network structure learning is often performed in a Bayesian setting, by evaluating candidate structures using their posterior probabilities for a given data set. Score-based algorithms then use those posterior probabilities as an objective function and return the maximum a posteriori network as the learned model. For discrete Bayesian networks, the canonical choice for a posterior score is the Bayesian Dirichlet equivalent uniform (BDeu) marginal likelihood with a uniform (U) graph prior (Heckerman et al., 1995). Its favourable theoretical properties descend from assuming a uniform prior both on the space of the network structures and on the space of the parameters of the network. In this paper, we revisit the limitations of these assumptions; and we introduce an alternative set of assumptions and the resulting score: the Bayesian Dirichlet sparse (BDs) empirical Bayes marginal likelihood with a marginal uniform (MU) graph prior. We evaluate its performance in an extensive simulation study, showing that MU+BDs is more accurate than U+BDeu both in learning the structure of the network and in predicting new observations, while not being computationally more complex to estimate.