Pith. sign in

REVIEW 3 major objections 4 minor 50 references

Scalable Structure Learning of Bayesian Networks by Learning Algorithm Ensembles

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Learning an ensemble of Bayesian-network structure learners and embedding it in a divide-and-conquer pipeline lifts accuracy on 10,000-variable networks by 30–225% over the best single-algorithm baseline.

desk verdict A useful portfolio-for-PEF result undermined by train/test family overlap and a theory-practice gap in selection; the 30-225% headline is in-distribution. read the letter →

arxiv 2506.22848 v1 pith:SK5UJEGH submitted 2025-06-28 cs.LG cs.AI

Shengcai Liu , Hui Ou-yang , Zhiyuan Wang , Cheng Chen , Qijun Cai , Yew-Soon Ong , Ke Tang This is my paper · ORCID
classification cs.LGcs.AI
keywords Bayesiannetworkstructurelearningdivide-and-conqueralgorithmensemblessubmodularoptimizationgreedyselectionlarge-scaleGaussiannetworksBICscorepartition-estimation-fusion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

At the core of this paper is a single claim: the main obstacle to learning very large Bayesian networks by divide-and-conquer is not the partition or fusion step but the instability of a single structure-learning algorithm across heterogeneous subproblems, and an automatically learned ensemble of algorithms removes that instability. The authors define a structure learning ensemble, a set of parameterized learner configurations, whose quality on any problem is the best quality among its members, and show that this quality function is monotone and submodular. They then propose Auto-SLE, a greedy constructor that repeatedly adds the configuration giving the largest marginal improvement and inherits a (1 - 1/e)-approximation guarantee relative to the optimal ensemble on the training problems. Embedded in the partition-estimation-fusion method, the learned ensemble (P/SLE) improves F1 and SHD accuracy on 10,000-variable Gaussian networks by roughly 30% to 225% over the best single-algorithm divide-and-conquer baseline and generalizes to 30,000-variable and non-block-structured networks with no retuning. If the paper is right, structure learning for large Bayesian networks ceases to be a contest between individual algorithms and becomes an automated portfolio-selection problem.

What carries the argument

The engine of the argument is the SLE learning problem and the greedy Auto-SLE algorithm. The quality measure is $Q(\mathcal{A},\mathcal{D}) = \max_{\theta \in \mathcal{A}} Q(\theta,\mathcal{D})$ averaged over training problems as $Q(\mathcal{A},\mathcal{T}) = \frac{1}{|\mathcal{T}|}\sum_{\mathcal{D}\in\mathcal{T}} Q(\mathcal{A},\mathcal{D})$, which is monotone and submodular by Fact 1 because the maximum operator has diminishing returns. Auto-SLE starts with an empty ensemble and, for $k$ iterations, adds the candidate configuration $\theta^\circ$ maximizing the marginal gain $\Delta(\theta|\mathcal{A}) = Q(\mathcal{A}\cup\{\theta\},\mathcal{T}) - Q(\mathcal{A},\mathcal{T})$, terminating early if no gain is positive; Theorem 1 gives the $(1-1/e)$ approximation with an explicit $k\epsilon$ term under imperfect parameter optimization. This learned ensemble is then dropped into the estimation step of PEF (partition, estimation, fusion), with the member whose output has the best BIC score selected for each subproblem.

What would settle it

Run P/SLE on 10,000-variable Gaussian problems constructed so that BIC systematically favors a low-F1 member, for example by varying the fGES penalty in ways that overfit BIC; if accuracy drops to the level of P/fGES on a set of such problems, the claimed ensemble gain is an artifact of BIC selection rather than of ensemble complementarity.

Watch

Extended reading notes

Core claim

The paper's central discovery is that a structure learning ensemble, learned once from diverse small training problems, can be transplanted into the estimation step of a divide-and-conquer method and keeps its accuracy advantage as the network scales to 10,000 and 30,000 variables. The learned ensemble contains only variants of a single score-based algorithm (fGES) with different penalty and maximum-parents settings; the constraint-based algorithm also in the design space never wins a slot. At training time the ensemble is scored by the best member's F1 on each problem, and at deployment the member output with the best BIC score is returned. The empirical core is a set of comparisons on ten benchmark-derived networks: P/SLE's F1-adjacent accuracy at 10,000 variables exceeds every single-algorithm divide-and-conquer baseline, typically by 30% or more and in one case by 133%, with corresponding structural-Hamming-distance reductions, and it is the only method that keeps accuracy roughly flat from 1,000 to 30,000 variables.

Load-bearing premise

The load-bearing premise is that the BIC score, used to select a member's output on new problems, picks the member with the highest true accuracy just as reliably as the oracle used when training the ensemble.

Editorial extensions

If this is right

  • At 10,000 variables, P/SLE reduces wall-clock time from no solution in 24 hours for fGES and PC-Stable to tens of minutes on most tested networks, while keeping F1-adjacent accuracy above 0.8 on most.
  • Accuracy improvements grow with network size: on 10,000-variable problems the F1-arrowhead improvement over the best baseline reaches 225%, and the margin continues to widen through 20,000 and 30,000 variables.
  • The learned ensemble transfers to networks not represented in training: small-world networks and a 4,441-node gene regulatory network, where P/SLE still beats every single-algorithm PEF baseline.
  • Auto-SLE's guarantee is not limited to Gaussian Bayesian networks; the monotone-submodular formulation and the greedy approximation hold for any quality measure, so the same pipeline could learn ensembles for other causal models.
  • The success of an all-fGES ensemble implies that the practical diversity that matters is in hyperparameter configurations, such as penalty and maximum parents, rather than in algorithm family differences.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A concrete test of the paper's logic would be to use a learned selection model, as the authors mention in future work, to pick the member algorithm from cheap problem features instead of BIC; if that matches or beats BIC selection, the ensemble method would become applicable to discrete and non-Gaussian Bayesian networks where BIC is a less reliable accuracy proxy.
  • The submodular portfolio view could be applied to the estimation step of other divide-and-conquer pipelines in causal discovery, such as subsampled or distributed skeleton finding, where per-subproblem algorithm instability is also the bottleneck.
  • Because the training and deployment sample size is fixed at 1000, the learned ensemble may be miscalibrated when deployment data are much denser or sparser; re-learning on training problems with matched sample size is an obvious extension the paper does not test.
  • The reported gains are relative to baselines that use default parameters; a more demanding comparison would pit P/SLE against single-algorithm PEF whose hyperparameters are tuned per network class by the same Bayesian optimizer, isolating the contribution of ensembling from that of parameter tuning.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces structure learning ensembles (SLEs) for divide-and-conquer Bayesian network structure learning. It formalizes SLE learning as maximizing average F1 over a training problem set with an oracle that selects the best member output, proves that this objective is monotone submodular, and provides a (1-1/e)-type approximation guarantee for the proposed greedy algorithm Auto-SLE. The learned SLE (which, in the experiments, consists of four fGES parameter configurations) is embedded into the PEF divide-and-conquer pipeline, yielding P/SLE. Experiments on bnlearn-based synthetic benchmarks with 1,000 and 10,000 variables report consistent accuracy improvements over single-algorithm baselines, with gains of 30%-225% at 10,000 variables; additional experiments on 20,000-30,000 variable versions of two networks, a small-world network, and a yeast gene network are used to support generalization claims.

Significance. If the central claims are established, the paper would make a useful practical contribution: automatically configured algorithm portfolios for large-scale BN structure learning, with an elegant and correct theoretical analysis of the training objective (Fact 1 and Theorem 1 follow from standard submodularity), open-sourced code, and extensive experiments comparing against strong baselines. The idea of using an ensemble in the estimation step of divide-and-conquer methods is sensible and underexplored. However, the significance is substantially tempered by two issues: the training set appears to be drawn from the same network families as the main test suite, which makes the headline improvements partly an in-distribution tuning effect, and the theoretical guarantee applies to an oracle-selection objective that is not what is used at deployment time (BIC-based selection). The out-of-distribution results (WS and Yeast, Table VII) are much weaker and sometimes negative, so the paper's generalization claims are not yet supported. The core method is defensible, but the current evidence base and framing need rework.

major comments (3)
  1. [Section IV.A.3 and Tables V-VI] The training set is generated by randomly drawing a base network from the same 32 bnlearn networks used to construct the 10 test networks, 100 times. Each of the 10 test networks therefore appears as a base topology in the training set with probability 1 - (31/32)^100 ≈ 96%. The simulated datasets differ, but the underlying DAG topology is shared, so the SLE's parameters (Table II) are effectively tuned to the exact structures on which the headline 30%-225% improvements (Table VI) are measured. The paper's statement that these training problems are 'independent of the testing problems' is therefore misleading. The only genuinely out-of-distribution tests (Table VII, WS and Yeast) show much smaller gains: for WS-1000 the F1 arrowhead improvement is -0.6%, and for Yeast the absolute F1 values are around 0.08-0.1. This does not support the abstract and introduction's claim of 'consistently achieving significantly higher accuracy' or generalization to 'different network characteristics'. A disjoint-family training/test split is needed to establish that the learned SLE provides more than benchmark-specific tuning.
  2. [Section III.D and Theorem 1] The training objective Q in Eq. (3) uses an oracle that selects the member algorithm with the best F1 score (ground truth known), while at test time Section III.D states that the best output is selected using BIC because ground truth is unavailable. Theorem 1's guarantee applies only to the oracle-based Q; it does not cover the BIC-selected output of the deployed system. The paper offers no analysis or empirical evidence that BIC-based selection tracks oracle F1 selection on the training problems or on the test problems. Since P/SLE's advantage over a single fGES configuration depends on BIC picking the right member, the mismatch between the optimized objective and the deployed selection rule is a load-bearing gap. This should be addressed, for example by training with a surrogate score that mimics test-time selection, or by reporting the correlation between BIC and F1 on training and test subproblems.
  3. [Table VI] The results on all 10,000-variable problems are based on a single generated problem per network, with no standard deviation or significance test. The improvement ratios of 30%-225% are computed against the best of P/PC-Stable and P/fGES, and the paper acknowledges that P/SLE(D) and P/fGES are identical because BIC always selects the fGES member. Thus the effective comparison set is two points per network, and the reported gains could be unstable across seeds. Given that the central quantitative claim rests on these numbers, the paper should provide multiple independent runs per network (or at least report the spread), particularly because the lower-variance 1,000-variable results in Table V show much smaller and sometimes negative improvements (e.g., Healthcare F1).
minor comments (4)
  1. [Section IV.A.2] There are several typos: 'conder' should be 'considered', 'he other' should be 'the other', and in Section IV.D 'structure leanring' should be 'structure learning'.
  2. [Table VII] The caption says 'P/SLE-B REPRESENTS THE BEST PERFORMANCE ACHIEVED AMONG P/SLE(R), P/SLE(D), P/FGES, AND P/PC-S TABLE', where 'P/PC-S TABLE' appears to be a typo for 'P/PC-Stable'.
  3. [Section IV.A.3] The paper says the SLE is learned from 'a diverse training problem set', but the learned SLE (Table II) contains only fGES with different parameter values; PC-Stable was never selected. This is a legitimate outcome, but the framing of an 'algorithm ensemble' combining multiple structure learning algorithms is somewhat stronger than what the experiments actually produce, and this should be stated more precisely.
  4. [Theorem 1 proof] The proof has minor notational issues: in Eq. (8) the line 'for all positive integers i < l≤ k' should be 'i < l ≤ k', and the telescoping sum notation is slightly compressed. The argument itself is sound.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the learning objective, submodularity proof, and greedy bound are self-contained, with only a minor non-load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained: Eq. (3) defines ensemble performance as a maximum over member algorithms, Eq. (5) proves monotone submodularity directly from that max definition, and Theorem 1 is the standard Nemhauser-Wolsey-Fisher approximation bound applied to the stated optimization objective. No fitted parameter is relabeled as a prediction, because the SLE members are selected on a training set with known ground truth while test-time F1 and SHD are measured on newly generated datasets, and the use of BIC for test-time selection (Section III.D) is an acknowledged oracle-to-deployment gap rather than a definitional identity. The only self-citation is the passing remark that generalization guarantees could be recovered using techniques from reference [36]; that remark is not load-bearing because no theorem from [36] is used and the generalization claim rests on experiments. The overlap between training and test bnlearn network families is a benchmark-design and generalization concern, not a circularity-by-construction issue, so it does not raise the circularity score.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on two categories of assumptions: standard mathematical facts (submodularity of maximum) that are proved in the paper, and domain choices (Gaussian linear BNs, training distribution overlap, and the BIC selection proxy) that are stated or implicit but not independently justified. The free parameters are the hand-chosen ensemble size, configuration ranges, training set, and stopping threshold. No new physical or model entities are introduced.

free parameters (4)
  • Ensemble size k = 4
    Chosen because further members improved training-set Q by less than 0.1.
  • Algorithm configuration ranges = PC-Stable: alpha in [0.01, 0.2], max depth [1, 1000]; fGES: lambda in [1.0, 1000.0], max parents [1, 1000]…
    Hand-specified design space; defines the candidate pool from which the SLE is learned.
  • Training set composition = 100 problems from 32 bnlearn networks, 5 to 1000 variables
    Hand-selected; defines the distribution assumed for generalization.
  • Termination threshold for Auto-SLE = 0.1 improvement in Q
    Stopping rule in Algorithm 1; a practical choice not justified by theory.
assumptions (4)
  • standard math Q(A,D) = max over theta in A of Q(theta,D) is monotone submodular
    Proved in Fact 1 using standard properties of maximum; underlies the greedy bound in Theorem 1.
  • domain assumption Data are generated from Gaussian Bayesian networks with linear structural equation models and i.i.d. observations
    Eq. (2) in Section II-A restricts the scope to Gaussian BNs with linear Gaussian conditionals.
  • domain assumption Test subproblems follow the same distribution as training problems
    Assumed in Section III-C for the transfer guarantee, and implicitly for the empirical generalization claims.
  • ad hoc to paper BIC-based selection at test time approximates oracle F1-based selection used in training
    No proof or analysis is given; the training objective (Eq. 3-4) uses Q=F1 with oracle, while Section III.D switches to BIC for deployment.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Scalable Structure Learning of Bayesian Networks by Learning Algorithm Ensembles." pith.science (2026). https://pith.science/paper/SK5UJEGH

@misc{pith2026250622848,
  author       = {Pith},
  title        = {Pith review of: Scalable Structure Learning of Bayesian Networks by Learning Algorithm Ensembles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SK5UJEGH}},
  note         = {Machine review of arXiv:2506.22848}
}
abstract

Learning the structure of Bayesian networks (BNs) from data is challenging, especially for datasets involving a large number of variables. The recently proposed divide-and-conquer (D\&D) strategies present a promising approach for learning large BNs. However, they still face a main issue of unstable learning accuracy across subproblems. In this work, we introduce the idea of employing structure learning ensemble (SLE), which combines multiple BN structure learning algorithms, to consistently achieve high learning accuracy. We further propose an automatic approach called Auto-SLE for learning near-optimal SLEs, addressing the challenge of manually designing high-quality SLEs. The learned SLE is then integrated into a D\&D method. Extensive experiments firmly show the superiority of our method over D\&D methods with single BN structure learning algorithm in learning large BNs, achieving accuracy improvement usually by 30\%$\sim$225\% on datasets involving 10,000 variables. Furthermore, our method generalizes well to datasets with many more (e.g., 30000) variables and different network characteristics than those present in the training data for learning the SLE. These results indicate the significant potential of employing (automatic learning of) SLEs for scalable BN structure learning.

Figures

Figures reproduced from arXiv: 2506.22848 by the authors.

Figure 1
Figure 1. An overview of P/SLE. The SLE is automatically learned by Auto [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Performance curves on Alarm and Asia problems with up to 30,000 variables. SHD and runtime are plotted on log scale. a-d are for Alarm problem, [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 47 canonical work pages

  1. [1]

    Bayesian networks: a model of self-activated memory for evidential reasoning,

    J. Pearl, “Bayesian networks: a model of self-activated memory for evidential reasoning,” in Proceedings of the 7th Conference of the Cognitive Science Society , 1985, pp. 329–334

    work page 1985

  2. [2]

    ——, Probabilistic reasoning in intelligent systems: networks of plau- sible inference, 1988

    work page 1988

  3. [3]

    A survey of community detection approaches: From statistical modeling to deep learning,

    D. Jin, Z. Yu, P. Jiao, S. Pan, D. He, J. Wu, P. S. Yu, and W. Zhang, “A survey of community detection approaches: From statistical modeling to deep learning,” IEEE Transactions on Knowledge and Data Engineering, vol. 35, no. 2, pp. 1149–1170, 2023

    work page 2023

  4. [4]

    A survey of bayesian network structure learning,

    N. K. Kitson, A. C. Constantinou, Z. Guo, Y . Liu, and K. Chobtham, “A survey of bayesian network structure learning,” Artificial Intelligence Review, pp. 1–94, 2023

    work page 2023

  5. [5]

    Order-independent constraint- based causal structure learning

    D. Colombo, M. H. Maathuis et al. , “Order-independent constraint- based causal structure learning.” Journal of Machine Learning Research, vol. 15, no. 1, pp. 3741–3782, 2014

    work page 2014

  6. [6]

    Ramsey, M

    J. Ramsey, M. Glymour, R. Sanchez-Romero, and C. Glymour, “A million variables and more: The fast greedy equivalence search algorithm for learning high-dimensional graphical causal models, with an applica- tion to functional magnetic resonance images,” International Journal of Data Science and Analytics , vol. 3, pp. 121–129, 2017

    work page 2017

  7. [7]

    The max-min hill- climbing bayesian network structure learning algorithm,

    I. Tsamardinos, L. E. Brown, and C. F. Aliferis, “The max-min hill- climbing bayesian network structure learning algorithm,”Machine learn- ing, vol. 65, pp. 31–78, 2006

    work page 2006

  8. [8]

    Learning bayesian networks with the bnlearn r package,

    M. Scutari, “Learning bayesian networks with the bnlearn r package,” Journal of Statistical Software , vol. 35, pp. 1–22, 2010

    work page 2010

Show all 50 references
  1. [9]

    Thps: Topolog- ical hawkes processes for learning causal structure on event sequences,

    R. Cai, S. Wu, J. Qiao, Z. Hao, K. Zhang, and X. Zhang, “Thps: Topolog- ical hawkes processes for learning causal structure on event sequences,” IEEE Transactions on Neural Networks and Learning Systems , 2022

    2022

  2. [10]

    Genenetweaver: in silico benchmark generation and performance profiling of network inference methods,

    T. Schaffter, D. Marbach, and D. Floreano, “Genenetweaver: in silico benchmark generation and performance profiling of network inference methods,” Bioinformatics, vol. 27, no. 16, pp. 2263–2270, 2011

    2011

  3. [11]

    Efficient and scalable structure learning for bayesian networks: Algorithms and applications,

    R. Zhu, A. Pfadler, Z. Wu, Y . Han, X. Yang, F. Ye, Z. Qian, J. Zhou, and B. Cui, “Efficient and scalable structure learning for bayesian networks: Algorithms and applications,” inProceedings of the 37th IEEE International Conference on Data Engineering , 2021, pp. 2613–2624

    2021

  4. [12]

    Large- scale hierarchical causal discovery via weak prior knowledge,

    X. Wang, T. Ban, L. Chen, D. Lyu, Q. Zhu, and H. Chen, “Large- scale hierarchical causal discovery via weak prior knowledge,” IEEE Transactions on Knowledge and Data Engineering , vol. 37, no. 5, pp. 2695–2711, 2025

    2025

  5. [13]

    Learning big gaussian bayesian networks: Partition, estimation and fusion,

    J. Gu and Q. Zhou, “Learning big gaussian bayesian networks: Partition, estimation and fusion,” Journal of Machine Learning Research , vol. 21, no. 1, pp. 6340–6370, 2020

    2020

  6. [14]

    Learning causal structures based on divide and conquer,

    H. Zhang, S. Zhou, C. Yan, J. Guan, X. Wang, J. Zhang, and J. Huan, “Learning causal structures based on divide and conquer,” IEEE Trans- actions on Cybernetics , vol. 52, no. 5, pp. 3232–3243, 2022

    2022

  7. [15]

    Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks,

    F. Croce and M. Hein, “Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks,” in Proceedings of the 37th International Conference on Machine Learning , 2020, pp. 2206–2216

    2020

  8. [16]

    Stochastic blockmodels: First steps,

    P. W. Holland, K. B. Laskey, and S. Leinhardt, “Stochastic blockmodels: First steps,” Social Networks, vol. 5, no. 2, pp. 109–137, 1983

    1983

  9. [17]

    Mixed membership stochastic blockmodels,

    E. M. Airoldi, D. Blei, S. Fienberg, and E. Xing, “Mixed membership stochastic blockmodels,” Advances in Neural Information Processing Systems, vol. 21, 2008

    2008

  10. [18]

    Exact recovery in the stochastic block model,

    E. Abbe, A. S. Bandeira, and G. Hall, “Exact recovery in the stochastic block model,” IEEE Transactions on Information Theory, vol. 62, no. 1, pp. 471–487, 2015

    2015

  11. [19]

    Large-sample learning of bayesian networks is np-hard,

    M. Chickering, D. Heckerman, and C. Meek, “Large-sample learning of bayesian networks is np-hard,” Journal of Machine Learning Research , vol. 5, pp. 1287–1330, 2004

    2004

  12. [20]

    Spirtes, C

    P. Spirtes, C. N. Glymour, and R. Scheines, Causation, Prediction, and Search. MIT Press, 2000

    2000

  13. [21]

    Time and sample efficient discovery of markov blankets and direct causal relations,

    I. Tsamardinos, C. F. Aliferis, and A. Statnikov, “Time and sample efficient discovery of markov blankets and direct causal relations,” in Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining , 2003, pp. 673–678

    2003

  14. [22]

    Bayesian belief networks: From construction to inference,

    R. R. Bouckaert, “Bayesian belief networks: From construction to inference,” Ph.D. dissertation, 1995

    1995

  15. [23]

    Learning bayesian network structures by searching for the best ordering with genetic algorithms,

    P. Larranaga, C. M. Kuijpers, R. H. Murga, and Y . Yurramendi, “Learning bayesian network structures by searching for the best ordering with genetic algorithms,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 26, no. 4, pp. 487–493, 1996

    1996

  16. [24]

    Parallel simulated annealing with a greedy algorithm for bayesian network structure learning,

    S. Lee and S. B. Kim, “Parallel simulated annealing with a greedy algorithm for bayesian network structure learning,” IEEE Transactions on Knowledge and Data Engineering , vol. 32, no. 6, pp. 1157–1166, 2020

    2020

  17. [25]

    Mutual information-guided GA for bayesian network structure learning,

    K. Yan, W. Fang, H. Lu, X. Zhang, J. Sun, and X. Wu, “Mutual information-guided GA for bayesian network structure learning,” IEEE Transactions on Knowledge and Data Engineering , vol. 35, no. 8, pp. 8282–8299, 2023

    2023

  18. [26]

    Learning equivalence classes of bayesian-network structures,

    D. M. Chickering, “Learning equivalence classes of bayesian-network structures,” Journal of Machine Learning Research, vol. 2, pp. 445–498, 2002

    2002

  19. [27]

    A new look at the statistical model identification,

    H. Akaike, “A new look at the statistical model identification,” IEEE Transactions on Automatic Control , vol. 19, no. 6, pp. 716–723, 1974

    1974

  20. [28]

    Estimating the dimension of a model,

    G. Schwarz, “Estimating the dimension of a model,” The Annals of Statistics, pp. 461–464, 1978

    1978

  21. [29]

    A bayesian method for the induction of probabilistic networks from data,

    G. F. Cooper and E. Herskovits, “A bayesian method for the induction of probabilistic networks from data,” Machine learning, vol. 9, pp. 309– 347, 1992

    1992

  22. [30]

    Dags with no tears: Continuous optimization for structure learning,

    X. Zheng, B. Aragam, P. K. Ravikumar, and E. P. Xing, “Dags with no tears: Continuous optimization for structure learning,” Advances in Neural Information Processing Systems , vol. 31, 2018

    2018

  23. [31]

    Smac3: A versatile bayesian optimization package for hyperparameter optimization,

    M. Lindauer, K. Eggensperger, M. Feurer, A. Biedenkapp, D. Deng, C. Benjamins, T. Ruhkopf, R. Sass, and F. Hutter, “Smac3: A versatile bayesian optimization package for hyperparameter optimization,” Jour- nal of Machine Learning Research, vol. 23, no. 1, pp. 2475–2483, 2022

    2022

  24. [32]

    An analysis of ap- proximations for maximizing submodular set functions - I,

    G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of ap- proximations for maximizing submodular set functions - I,” Mathmatical Programming, vol. 14, no. 1, pp. 265–294, 1978

    1978

  25. [33]

    Automatic construction of parallel portfolios via explicit instance grouping,

    S. Liu, K. Tang, and X. Yao, “Automatic construction of parallel portfolios via explicit instance grouping,” in Proceedings of the 33rd AAAI Conference on Artificial Intelligence, AAAI’2019 , Honolulu, HI, Jan 2019, pp. 1560–1567

    2019

  26. [34]

    How good is neural combina- torial optimization? A systematic evaluation on the traveling salesman problem,

    S. Liu, Y . Zhang, K. Tang, and X. Yao, “How good is neural combina- torial optimization? A systematic evaluation on the traveling salesman problem,” IEEE Computational Intelligence Magazine , vol. 18, no. 3, pp. 14–28, 2023

    2023

  27. [35]

    Learn to optimize—a brief overview,

    K. Tang and X. Yao, “Learn to optimize—a brief overview,” National Science Review, vol. 11, no. 8, p. nwae132, 2024

    2024

  28. [36]

    On performance estimation in automatic algorithm configuration,

    S. Liu, K. Tang, Y . Lei, and X. Yao, “On performance estimation in automatic algorithm configuration,” in Proceedings of the 34th AAAI Conference on Artificial Intelligence, AAAI’2020 , New York, NY , Feb 2020, pp. 2384–2391

    2020

  29. [37]

    Tetrad — a toolbox for causal discovery,

    J. D. Ramsey, K. Zhang, M. Glymour, R. S. Romero, B. Huang, I. Ebert- Uphoff, S. Samarasinghe, E. A. Barnes, and C. Glymour, “Tetrad — a toolbox for causal discovery,” in Proceedings of the 8th International Workshop on Climate Informatics , 2018

    2018

  30. [38]

    Concave penalized estimation of sparse gaussian bayesian networks,

    B. Aragam and Q. Zhou, “Concave penalized estimation of sparse gaussian bayesian networks,” Journal of Machine Learning Research , vol. 16, no. 1, pp. 2273–2328, 2015

    2015

  31. [39]

    Learning bayesian network model structure from data,

    D. Margaritis et al., “Learning bayesian network model structure from data,” Ph.D. dissertation, School of Computer Science, Carnegie Mellon University Pittsburgh, PA, USA, 2003

    2003

  32. [40]

    Algorithms for large scale markov blanket discovery

    I. Tsamardinos, C. F. Aliferis, A. R. Statnikov, and E. Statnikov, “Algorithms for large scale markov blanket discovery.” in FLAIRS conference, vol. 2. St. Augustine, FL, 2003, pp. 376–380

    2003

  33. [41]

    Learning gaussian graphical models of gene networks with false discovery rate control,

    J. M. Pena, “Learning gaussian graphical models of gene networks with false discovery rate control,” in European conference on evolutionary computation, machine learning and data mining in bioinformatics . Springer, 2008, pp. 165–176. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO....

    2008

  34. [42]

    Speculative markov blanket discovery for optimal feature selection,

    S. Yaramakala and D. Margaritis, “Speculative markov blanket discovery for optimal feature selection,” in Fifth IEEE International Conference on Data Mining (ICDM’05) . IEEE, 2005, pp. 4–pp

    2005

  35. [43]

    Learning bayesian network structure from massive datasets: The

    N. Friedman, I. Nachman, and D. Pe’er, “Learning bayesian network structure from massive datasets: The” sparse candidate” algorithm,” arXiv preprint arXiv:1301.6696 , 2013

    2013 arXiv

  36. [44]

    A hybrid algorithm for bayesian network structure learning with application to multi-label learning,

    M. Gasse, A. Aussem, and H. Elghazel, “A hybrid algorithm for bayesian network structure learning with application to multi-label learning,” Expert Systems with Applications, vol. 41, no. 15, pp. 6755–6772, 2014

    2014

  37. [45]

    On the evolution of random graphs,

    P. Erd ˝os, A. R ´enyi et al. , “On the evolution of random graphs,” Publ. math. inst. hung. acad. sci , vol. 5, no. 1, pp. 17–60, 1960

    1960

  38. [46]

    Scale-free networks,

    A.-L. Barab ´asi and E. Bonabeau, “Scale-free networks,” Scientific amer- ican, vol. 288, no. 5, pp. 60–69, 2003

    2003

  39. [47]

    Maximal prime subgraph decompo- sition of bayesian networks,

    K. G. Olesen and A. L. Madsen, “Maximal prime subgraph decompo- sition of bayesian networks,” IEEE Transactions on Systems, Man, and Cybernetics. Part B, vol. 32, no. 1, pp. 21–31, 2002

    2002

  40. [48]

    The igraph software package for complex network research,

    G. Csardi, T. Nepusz et al., “The igraph software package for complex network research,” InterJournal, complex systems , vol. 1695, no. 5, pp. 1–9, 2006

    2006

  41. [49]

    Collective dynamics of ‘small- world’networks,

    D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small- world’networks,” nature, vol. 393, no. 6684, pp. 440–442, 1998

    1998

  42. [50]

    Towards feature-free tsp solver selection: A deep learning approach,

    K. Zhao, S. Liu, J. X. Yu, and Y . Rong, “Towards feature-free tsp solver selection: A deep learning approach,” in Proceedings of the 2021 International Joint Conference on Neural Networks, IJCNN’2021 , Virtual Event, Jul 2021

    2021

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.