Learning Bayesian Networks with the bnlearn R Package

Marco Scutari

article2009Journal of Statistical Software (2010), 35(3), 1-222,030 citations

Presents the bnlearn R package, providing a unified and computationally scalable framework for learning the structure of discrete and continuous Bayesian networks through constraint-based and score-based algorithms.

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Modern data analytics across healthcare, genetics, and operational monitoring increasingly depend on understanding complex probabilistic relationships within high-dimensional datasets. Bayesian networks represent these interactions as directed acyclic graphs, mapping direct dependencies while breaking down complex global probability distributions into manageable local calculations. However, identifying the true underlying network structure from massive datasets is computationally intensive, and existing software implementations typically restrict analysts by tying specific learning algorithms to rigid statistical tests or data formats.

The article demonstrates the implementation and capabilities of bnlearn, a software package developed for the R environment that provides a flexible, highly optimized framework for learning Bayesian network structures from both discrete and continuous data. The author set out to evaluate whether decoupling learning algorithms from underlying statistical metrics, combined with algorithmic enhancements and parallel execution, delivers a more versatile and scalable solution for network structure learning.

To evaluate the system, the article conducts comparative analyses on both synthetic and empirical benchmarks, including a 37-variable medical monitoring benchmark containing 20,000 observations (the ALARM network) and an educational dataset of 88 students evaluated across five continuous examination subjects. The evaluation covers both constraint-based approaches—which map networks using conditional independence tests—and score-based greedy search algorithms. The package incorporates computational optimizations, such as backtracking to reduce redundant statistical testing by roughly half, caching to speed score comparisons, and parallel computing capabilities.

The findings show that constraint-based algorithms (including Grow-Shrink and variants of the Incremental Association algorithm) and greedy hill-climbing consistently recover the true structural relationships within a few arcs. On the ALARM dataset, the default optimized constraint-based algorithms completed execution in approximately 13 to 19 seconds, while hill-climbing required roughly 72 seconds. Crucially, the choice of statistical metric heavily impacts accuracy: employing Monte Carlo permutation tests on sparse discrete data reduced structural learning errors from twelve missed connections under standard parametric tests down to only five. Furthermore, the analysis shows that integrating prior domain expertise through whitelisting or blacklisting specific connections effectively resolves structural ambiguities and optimizes network orientation.

These results demonstrate that analysts can achieve substantial performance gains and lower computational costs without sacrificing model accuracy. The capacity to mix arbitrary statistical criteria with different learning algorithms allows teams to tailor models directly to the mathematical properties of their data, mitigating risks of misidentification in sparse or high-dimensional environments. For enterprise and research workflows, this flexibility reduces development cycle times while supporting rigorous causal inference and predictive modeling.

Organizations analyzing complex multivariate data should adopt flexible workflows that decouple network search algorithms from independence tests, prioritizing permutation-based tests when dealing with small sample sizes or sparse contingency tables. In addition, practitioners should systematically incorporate known operational constraints and expert rules into the model configuration to guide edge direction and eliminate spurious dependencies. Users should note that while the package provides advanced structural learning and diagnostics, the current implementation handles discrete and continuous datasets separately under multinomial or multivariate normal assumptions, rather than mixed hybrid networks within a single model.

University of Padua bayesian network structure parallel computing network architecture Bayesian networks bnlearn R package network structure learning

arXiv: 0908.3817 cran/bnlearn

Paper: Learning Bayesian networks: The combination of knowledge and statistical data, D. Heckerman et al. (1994). It derives the foundational Bayesian Dirichlet scoring criteria (such as the BDe metric) and search strategies for structure learning that bnlearn implements for score-based network discovery.
Paper: A Bayesian method for the induction of probabilistic networks from data, G. Cooper et al. (1992). It introduces the landmark K2 score and greedy heuristic algorithm for learning Bayesian network structures from data, which form core reference implementations in bnlearn.
Paper: A Tutorial on Learning with Bayesian Networks, David Heckerman (1999). It provides an essential tutorial on probabilistic graphical representations, parameter updating, and structure learning principles that underpin the bnlearn package.
Paper: Bayesian Network Classifiers, NIR FRIEDMAN et al. (1997). It formalizes Bayesian network classifier structures, such as Tree-Augmented Naive Bayes (TAN) and score-based learning via description length criteria, supported in bnlearn.
Paper: An Introduction to Variational Methods for Graphical Models, MICHAEL I. JORDAN et al. (1999). It establishes foundational theory on probabilistic graphical models and inference algorithms essential for reasoning over learned Bayesian network structures.

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Cover for Learning Bayesian Networks with the bnlearn R Package

Abstract

bnlearn is an R package which includes several algorithms for learning the structure of Bayesian networks with either discrete or continuous variables. Both constraint-based and score-based algorithms are implemented, and can use the functionality provided by the snow package to improve their performance via parallel computing. Several network scores and conditional independence algorithms are available for both the learning algorithms and independent use. Advanced plotting options are provided by the Rgraphviz package.

Topics

bayesian network structure parallel computing network architecture

Keywords

Bayesian networks bnlearn R package network structure learning Markov blanket discovery constraint-based learning algorithms score-based algorithms graphical models

1 Introduction
2 Bayesian networks
3 Structure learning algorithms
4 Package implementation
4.1 Structure learning algorithms
4.2 Conditional independence tests
4.3 Network scores
4.4 Arc whitelisting and blacklisting
5 A simple example
5.1 Loading the package
5.2 Learning a Bayesian network from data
5.3 Network analysis and manipulation
5.4 Debugging utilities and diagnostics
6 Practical examples
6.1 The ALARM network
6.2 The examination marks data set
7 Other packages for learning Bayesian networks
8 Conclusions
References

Knowls

Knowl 1 — Structure Learning Algorithm Suite and Execution Modes in bnlearn
model/method
The bnlearn package provides several structure learning algorithms for Bayesian networks with discrete or continuous variables, along with multiple algorithmic execution modes:
1. Constraint-Based Algorithms:
  - Grow-Shrink (gs): Learns the network structure based on the Grow-Shrink Markov blanket detection algorithm by performing a forward inclusion phase followed by a backward false-positive removal phase.
  - Incremental Association (iamb): Learns the Markov blanket via a two-phase scheme (forward selection followed by false-positive elimination) to reconstruct the graph skeleton.
  - Fast Incremental Association (fast.iamb): A variant of IAMB that employs speculative stepwise forward selection to reduce the total number of conditional independence tests.
  - Interleaved Incremental Association (inter.iamb): A variant of IAMB that interleaves forward selection and backward elimination steps to prevent false positives during Markov blanket estimation.
  - Max-Min Parents and Children (mmpc): A forward selection algorithm for neighborhood detection based on maximizing the minimum association measure observed with any subset of previously selected variables; it learns the undirected skeleton of the Bayesian network without orienting edges.
2. Score-Based Algorithms:
  - Hill-Climbing (hc): A greedy local search across the directed acyclic graph (DAG) space. It supports random restarts (restart), network perturbation operations (perturb), and preseeded initial structures (start) to escape local score maxima.
3. Implementations and Optimizations:
  - Optimized (Default): Uses backtracking across node pairs in constraint-based learning to roughly halve the required number of independence tests. In score-based learning, it utilizes score caching, score decomposability, and score equivalence to avoid redundant score evaluations.
  - Unoptimized: Faithfully runs the textbook algorithm without caching or backtracking, useful for benchmarking.
  - Parallel: Uses the snow R package to parallelize conditional independence tests across multi-core machines or clusters.
Knowl 2 — Discrete Conditional Independence Tests
equation
For discrete random variables $X$ and $Y$ given a conditioning set of discrete variables $Z$ , conditional independence tests evaluate the 3-dimensional contingency table of observed cell counts $\{n_{ijk}\}$ for $i = 1, \dots, R$ (levels of $X$ ), $j = 1, \dots, C$ (levels of $Y$ ), and $k = 1, \dots, L$ (configurations of $Z$ ), where $n = \sum_{i,j,k} n_{ijk}$ is the sample size, $n_{i+k} = \sum_{j=1}^C n_{ijk}$ , $n_{+jk} = \sum_{i=1}^R n_{ijk}$ , and $n_{++k} = \sum_{i=1}^R \sum_{j=1}^C n_{ijk}$ .
- Mutual Information Test (mi, mc-mi): $MI(X, Y \mid Z) = \sum_{i=1}^{R} \sum_{j=1}^{C} \sum_{k=1}^{L} \frac{n_{ijk}}{n} \log \left( \frac{n_{ijk} n_{++k}}{n_{i+k} n_{+jk}} \right)$ The statistic $2n \cdot MI(X, Y \mid Z)$ is proportional to the log-likelihood ratio test statistic $G^2$ and deviance. Under the null hypothesis of conditional independence, it asymptotically follows a $\chi^2$ distribution with $(R-1)(C-1)L$ degrees of freedom (mi), or can be tested nonparametrically via Monte Carlo permutations (mc-mi).
- Pearson's $X^2$ Test (x2, mc-x2): $X^2(X, Y \mid Z) = \sum_{i=1}^{R} \sum_{j=1}^{C} \sum_{k=1}^{L} \frac{(n_{ijk} - m_{ijk})^2}{m_{ijk}}, \quad \text{where } m_{ijk} = \frac{n_{i+k} n_{+jk}}{n_{++k}}$ Evaluated against an asymptotic $\chi^2$ distribution with $(R-1)(C-1)L$ degrees of freedom (x2) or via Monte Carlo permutation testing (mc-x2).
- Fast Mutual Information Heuristic (fmi): Computes $MI(X, Y \mid Z)$ as above, but truncates the statistic to zero if $\frac{n}{(R-1)(C-1)L} < 5$ , avoiding unreliable tests in sparse contingency tables.
- AIC-Based Independence Test (aict): Rejects the null hypothesis of conditional independence when: $MI(X, Y \mid Z) > \frac{(R - 1)(C - 1)L}{n}$ which corresponds to an increase in the global network AIC score upon adding an edge between $X$ and $Y$ .
Knowl 3 — Continuous Conditional Independence Tests Based on Partial Correlation
equation
For continuous random variables $X$ and $Y$ conditioned on a subset of continuous variables $Z$ under the assumption of multivariate normality, conditional independence tests evaluate functions of the sample partial correlation coefficient $\rho_{XY \mid Z}$ :
- Linear Correlation Student's $t$ Test (cor, mc-cor): Tests $\rho_{XY \mid Z} = 0$ using Student's $t$ distribution with $n - |Z| - 2$ degrees of freedom (cor) or a Monte Carlo permutation test (mc-cor), where $n$ is sample size and $|Z|$ is the cardinality of $Z$ .
- Fisher's $Z$ Test (zf, mc-zf): Applies Fisher's transformation to the partial correlation: $Z(X, Y \mid Z) = \frac{1}{2} \sqrt{n - |Z| - 3} \log \left( \frac{1 + \rho_{XY \mid Z}}{1 - \rho_{XY \mid Z}} \right)$ Under the null hypothesis of conditional independence, $Z(X, Y \mid Z)$ asymptotically follows a standard normal distribution $\mathcal{N}(0, 1)$ (zf), or can be tested via Monte Carlo permutation (mc-zf).
- Continuous Mutual Information Test (mi-g, mc-mi-g): $MI_g(X, Y \mid Z) = -\frac{1}{2} \log \left( 1 - \rho_{XY \mid Z}^2 \right)$ The scaled statistic $2n \cdot MI_g(X, Y \mid Z)$ is asymptotically distributed as $\chi^2$ with 1 degree of freedom (mi-g) or evaluated via Monte Carlo permutation (mc-mi-g).
Knowl 4 — Network Scoring Functions for Discrete and Continuous Bayesian Networks
equation
For a Bayesian network structure $G$ on variables $\mathbf{V} = \{X_1, \dots, X_v\}$ , score-based learning algorithms evaluate candidate graphs using discrete or continuous network scores:
1. Discrete Scores:
  - Log-Likelihood (loglik) and Likelihood (lik): $\log L(X_1, \dots, X_v) = \sum_{i=1}^v \sum_{j=1}^{L_i} \sum_{k=1}^{R_i} n_{ijk} \log \left( \frac{n_{ijk}}{\sum_{k'=1}^{R_i} n_{ijk'}} \right)$ where $R_i$ is the number of states of node $X_i$ , $L_i$ is the number of parent configurations of $X_i$ , and $n_{ijk}$ is the count of observations where $X_i = k$ and its parents $\Pi_{X_i}$ are in state $j$ .
  - Akaike Information Criterion (aic) and Bayesian Information Criterion (bic): $AIC = \log L(X_1, \dots, X_v) - d, \qquad BIC = \log L(X_1, \dots, X_v) - \frac{d}{2} \log n$ where $d = \sum_{i=1}^v (R_i - 1) L_i$ is the number of free parameters in the network and $n$ is the total sample size. BIC is equivalent to the Minimum Description Length (MDL) score. Both AIC and BIC are score-equivalent.
  - Bayesian Dirichlet Equivalent Score (bde): The logarithm of the Dirichlet posterior density under uniform prior hyperparameters, satisfying score equivalence across Markov equivalent DAGs.
  - K2 Score (k2): The logarithm of the Dirichlet posterior density with uniform parameter priors: $K2 = \prod_{i=1}^{v} K2(X_i), \quad K2(X_i) = \prod_{j=1}^{L_i} \frac{(R_i - 1)!}{\left( \sum_{k=1}^{R_i} n_{ijk} + R_i - 1 \right)!} \prod_{k=1}^{R_i} n_{ijk}!$ Unlike bde, aic, and bic, the K2 score is not score-equivalent.
2. Continuous Score:
  - Bayesian Gaussian Equivalent Score (bge): A score-equivalent Gaussian posterior density assuming a Wishart prior distribution for continuous multivariate normal networks.
Knowl 5 — Arc Whitelisting and Blacklisting Semantics
model/method
In bnlearn, prior structural constraints can be enforced across all learning algorithms via the whitelist and blacklist arguments, which take sets of directed arc pairs $(A, B)$ representing $A \to B$ :
- Bidirectional Whitelist: If both $A \to B$ and $B \to A$ are whitelisted, an edge connecting $A$ and $B$ must exist in the learned network, but its orientation ( $A \to B$ , $B \to A$ , or undirected $A - B$ ) is left to the learning algorithm.
- Unidirectional Whitelist: If $A \to B$ is whitelisted but $B \to A$ is not, the directed arc $A \to B$ is strictly enforced in the learned graph. This automatically blacklists both the reverse arc $B \to A$ and the undirected edge $A - B$ .
- Bidirectional Blacklist: If both $A \to B$ and $B \to A$ are blacklisted, no edge between $A$ and $B$ can appear in the network, precluding $A \to B$ , $B \to A$ , and $A - B$ .
- Unidirectional Blacklist: If $A \to B$ is blacklisted but $B \to A$ is not, the directed arc $A \to B$ and the undirected edge $A - B$ are prohibited, but the reverse arc $B \to A$ remains eligible for inclusion.
- Precedence Rule: If an arc is specified in both whitelist and blacklist, the whitelist takes precedence and the arc is removed from the blacklist.

Knowl 6 — Structure Learning Performance on the ALARM Network

data/table

The ALARM network benchmark evaluates constraint-based and score-based learning algorithms on a sample of $n = 20{,}000$ observations generated from the true ALARM network ( $37$ discrete variables with 2 to 4 levels each, and $46$ true arcs). Benchmarks were executed on an Intel Core 2 Duo machine with 1GB of RAM.

Metric	gs	iamb	fast.iamb	inter.iamb	hc
Independence tests / Network comparisons	1727	2874	2398	3106	2841
Learned arcs (directed / undirected)	42 (29/13)	43 (29/14)	45 (32/13)	43 (30/13)	53 (53/0)
Execution time (seconds)	13.54360	17.63735	14.80890	18.59825	72.38705

All constraint-based algorithms (gs, iamb, fast.iamb, inter.iamb) successfully reconstruct the ALARM network within a few arcs and directions, requiring roughly $1{,}700$ to $3{,}100$ conditional independence tests and executing in 13.5–18.6 seconds. gs requires the fewest tests (1727) and has the lowest runtime (13.54 s). Hill-climbing (hc with BIC) performs 2841 score comparisons, completely directs all 53 learned arcs (including several extraneous arcs beyond the true 46), and requires substantially longer computation time (72.39 s).

Knowl 7 — Accuracy Gains of Monte Carlo Permutation Tests on Sparse Discrete Data
empirical result
When learning discrete Bayesian network structures from data with high-cardinality conditioning sets, the resulting conditional contingency tables frequently suffer from severe sparsity, causing asymptotic $\chi^2$ distribution assumptions to break down.

In structure learning on the 37-variable ALARM network data ( $n = 20{,}000$ ) using the Grow-Shrink (gs) algorithm:
- Using the asymptotic parametric Pearson's $X^2$ test (test = "x2") results in $12$ true network arcs being missed.
- Using the Monte Carlo permutation Pearson's $X^2$ test (test = "mc-x2" with $B = 10{,}000$ permutations) misses only $5$ true arcs.
Nonparametric permutation tests substantially improve edge recovery accuracy over parametric asymptotic tests in discrete Bayesian networks when data sparsity violates asymptotic assumptions.
Knowl 8 — Continuous Structure Learning and Conditional Independence on Examination Marks Data
empirical result
Structure learning on the continuous marks dataset ( $n = 88$ student scores across Mechanics MECH, Vectors VECT, Algebra ALG, Analysis ANL, and Statistics STAT) demonstrates consistency across different continuous conditional independence tests and structure learning algorithms under multivariate normality:
1. Conditional Independence Tests: Testing the conditional independence of $(\text{MECH}, \text{VECT})$ and $(\text{ANL}, \text{STAT})$ given $\text{ALG}$ yields non-significant $p$ -values across all parametric and nonparametric test formulations:
  - $\text{MECH} \perp \!\!\! \perp \text{ANL} \mid \text{ALG}$ : Student's $t$ linear correlation test yields $\rho = 0.0352$ ( $df = 85$ , $p = 0.7459$ ).
  - $\text{STAT} \perp \!\!\! \perp \text{VECT} \mid \text{ALG}$ $STAT ⊥ ⊥ VECT ∣ ALG$ :
    - Student's $t$ linear correlation (cor): $\rho = 0.0527$ ( $df = 85$ , $p = 0.628$ ).
    - Fisher's $Z$ asymptotic test (zf): $p = 0.6289$ .
    - Monte Carlo permutation linear correlation (mc-cor): $p = 0.6332$ .
    - Continuous mutual information asymptotic test (mi-g): $p = 0.6209$ .
    - Continuous mutual information Monte Carlo test (mc-mi-g): $p = 0.6226$ .
2. Learned Network Structures: All constraint-based algorithms (gs, iamb, fast.iamb, inter.iamb), neighborhood discovery (mmpc), and score-based search (hc) learn consistent graphical models showing that ANL and STAT are conditionally independent of MECH and VECT given ALG, identifying algebra as the central bridging skill across the examination subjects.

Coverage note — Standard theoretical definitions of Bayesian network factorizations (Pearl 1988, Korb and Nicholson 2004), the generic Inductive Causation (IC) steps, and basic syntax for R helper functions (such as `empty.graph` and `modelstring`) were omitted as standard background and software syntactic wrappers.

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Citation

MLA

Scutari, M. “Learning Bayesian Networks with the Bnlearn R Package”. Journal of Statistical Software (2010), 35(3), 1-22, 2009, http://arxiv.org/abs/0908.3817v2.

APA

Scutari, M. (2009). Learning Bayesian Networks with the bnlearn R Package. Journal of Statistical Software (2010), 35(3), 1-22. http://arxiv.org/abs/0908.3817v2

Chicago

Scutari, M. 2009. “Learning Bayesian Networks with the Bnlearn R Package”. Journal of Statistical Software (2010), 35(3), 1-22. http://arxiv.org/abs/0908.3817v2.

Harvard

Scutari, M. (2009) “Learning Bayesian Networks with the bnlearn R Package”, Journal of Statistical Software (2010), 35(3), 1-22 [Preprint]. Available at: http://arxiv.org/abs/0908.3817v2.

Vancouver

1. Scutari M (2009) Learning Bayesian Networks with the bnlearn R Package. Journal of Statistical Software (2010), 35(3), 1-22

BibTeX

@article{scutari2009learning,
  title = {Learning Bayesian Networks with the bnlearn R Package},
  author = {Scutari, Marco},
  year = {2009},
  journal = {Journal of Statistical Software (2010), 35(3), 1-22},
  url = {http://arxiv.org/abs/0908.3817v2},
  eprint = {0908.3817}
}

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Learning Bayesian Networks with the bnlearn R Package

Abstract

Topics

Keywords

Table of Contents

Knowls

Knowl 1 — Structure Learning Algorithm Suite and Execution Modes in bnlearn

Knowl 2 — Discrete Conditional Independence Tests

Knowl 3 — Continuous Conditional Independence Tests Based on Partial Correlation

Knowl 4 — Network Scoring Functions for Discrete and Continuous Bayesian Networks

Knowl 5 — Arc Whitelisting and Blacklisting Semantics