# A Theoretical Analysis of the BDeu Scores in Bayesian Network Structure Learning

arXiv: https://arxiv.org/abs/1607.04427 · HTML: https://mlsanity.com/paper/1607.04427/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2016.

## Built on by (most cited first, 5 of 5)

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- [Quotient Normalized Maximum Likelihood Criterion for Learning Bayesian Network Structures](https://mlsanity.com/paper/2408.14935/) (2024)
- [Exact Learning Augmented Naive Bayes Classifier](https://mlsanity.com/paper/2107.03018/) (2021)

## Abstract (arXiv)

In Bayesian network structure learning (BNSL), we need the prior probability over structures and parameters. If the former is the uniform distribution, the latter determines the correctness of BNSL. In this paper, we compare BDeu (Bayesian Dirichlet equivalent uniform) and Jeffreys' prior w.r.t. their consistency. When we seek a parent set U of a variable X, we require regularity that if H(X|U)≤ H(X|U') and Usubsetneq U', then U should be chosen rather than U'. We prove that the BDeu scores violate the property and cause fatal situations in BNSL. This is because for the BDeu scores, for any sample size n,there exists a probability in the form P(X,Y,Z)=P(XZ)P(YZ)/P(Z) such that the probability of deciding that X and Y are not conditionally independent given Z is more than a half. For Jeffreys' prior, the false-positive probability uniformly converges to zero without depending on any parameter values, and no such an inconvenience occurs.