DAGGuard

Exact and greedy BIC refinement of learned directed acyclic graphs

DAGGuard is a post-learning refinement method for an already oriented candidate DAG. Its fixed-candidate objective is defined conditional on the supplied data and graph: DAGGuard only decides which candidate edges to retain and never adds an omitted edge or reverses an upstream orientation. In the accompanying paper, all end-to-end experiments and the commercial application use NOTEARS-generated candidates; finite-sample performance after other upstream DAG learners has not been established here.

The accompanying manuscript is “DAGGuard: Exact and Greedy BIC Refinement of Learned DAGs with an Application to Commercial Swine Production.”

Why DAGGuard?

A learned DAG can be overconnected or contain weakly supported edges. DAGGuard separates candidate generation from post-learning assessment:

DAG learning (NOTEARS in this study) -> fixed candidate DAG -> DAGGuard -> refined DAG

For Gaussian local BIC,

BIC_j(S) = n log(RSS_j(S)/n) + (|S| + 1) log(n)

and the deletion-subgraph problem separates exactly by child. DAGGuard therefore provides two complementary algorithms:

• DAGGuard-Greedy: fast repeated best single-edge deletion.
• DAGGuard-Exact: exact child-wise best-subset selection using enumeration and branch-and-bound, with an explicit numerical optimality flag.

The one-edge BIC rule has the exact partial-R-squared interpretation

delete edge iff partial R^2 < 1 - n^(-1/n).

DAGGuard is a score-based refinement procedure, not a finite-sample nominal FDR method. Its formal guarantees are conditional on a fixed or independently learned candidate. The current end-to-end empirical validation is specifically for NOTEARS-generated candidates.

For conventional Gaussian BIC, every public refinement entry point requires each child's centered full candidate-parent design to have full column rank, q_j < n - 1, and strictly positive, numerically nondegenerate full-model residual variance. Rank-deficient, saturated, constant-response, and degenerate candidate regressions are rejected with informative exceptions. Before numerical rank assessment, nonconstant candidate-parent columns are normalized by their Euclidean norms, so validation is stable to changes of measurement units. globally_optimal=True means the exact search established the minimum score within the documented numerical tolerance and did not hit its search limit; it does not imply a unique representative when multiple subsets are numerically tied.

Installation

Python 3.12 was used for the reported analyses. Core package versions are pinned in requirements.txt.

python -m pip install -r requirements.txt

Public API

New code should import from dagguard.py:

from dagguard import dagguard_exact, dagguard_greedy, pruning_pressure

exact = dagguard_exact(X, candidate)
greedy = dagguard_greedy(X, candidate)

print(exact.globally_optimal)
print(exact.total_bic)

The lower-level names exact_refine_dag and greedy_refine_dag exported by dagguard.py use the same validation policy. local_bic_refinement.py remains the tested numerical engine for backward compatibility with the earlier reproducibility commit.

Run the minimal example:

python -m examples.dagguard_quickstart

Main empirical results

Controlled fixed-candidate experiments

Across 1,200 controlled candidate-contamination experiments, DAGGuard removed nearly all added false positives when the candidate was an overconnected screen. Under 100% added false-positive contamination, candidate FDR was 0.500 and DAGGuard-Exact FDR was 0.013 with TPR 1.000. DAGGuard-Greedy and DAGGuard-Exact returned the same edge set in 1,192/1,200 replicates.

Seven-method end-to-end benchmark

Across 240 common Gaussian, centered-exponential, and centered-Gumbel simulation datasets using NOTEARS candidates for the DAGGuard rows, the pooled skeleton false-discovery proportion decreased from 0.148 for NOTEARS to 0.066 for DAGGuard-Exact while pooled TPR changed only from 0.860 to 0.856. Benefits were concentrated in regimes where NOTEARS over-selected. The repository includes transparent comparator code and provenance for Wang et al. (2026), Li & Wang PC-FDR, PC-p, and ordinary PC.

Primary summary: results/seven_method_benchmark/simulation_primary_seven_methods.csv.

NOTEARS tuning sensitivity

A 3-by-3 grid varied the NOTEARS L1 penalty and post-estimation threshold. Across all evaluated dimension-by-tuning cells, DAGGuard-Exact reduced mean FDR and SHD with comparatively small TPR changes. The d=10 cells use 120 datasets each; the targeted d=20 strong-signal cells use 15 datasets each and are treated as exploratory. Results are in results/notears_tuning_sensitivity/.

Commercial swine application

The authorized analysis uses 2,556 complete lots and 37 variables. The pinned NOTEARS candidate has 185 edges; DAGGuard-Exact retains 87 and DAGGuard-Greedy 89. PRRS is adjacent to 60-day nursery mortality in all seven benchmark methods, while MYCO and third-quarter placement are supported by six. Four outgoing NOTEARS mortality relationships are removed by both DAGGuard variants and are absent from all four comparator graphs.

The proprietary row-level data are not distributed. The real-data workflow records a SHA256 hash and exports only non-row-level summaries.

Repository map

• dagguard.py - validated public DAGGuard API.
• local_bic_refinement.py - backward-compatible numerical engine: local BIC, exact search, greedy search, pruning pressure, and graph metrics.
• candidate_contamination_simulations.py - 12-setting fixed-candidate experiment.
• reproduce_simulations.py - primary NOTEARS simulation workflow.
• additional_noise_sensitivity.py - Gaussian/exponential/Gumbel sensitivity analysis.
• notears_tuning_sensitivity.py - NOTEARS penalty/threshold sensitivity analysis.
• realdata_postselection_diagnostics.py - authorized swine-data analysis.
• synthetic_application_twin.py - public 37-variable workflow without proprietary observations.
• reproduce_submission.sh - staged reproduction entry point.
• benchmarks/seven_method/ - audited competitor implementations and real-data benchmark runner.
• benchmarks/seven_method/SOURCE_AUDIT.md - source-by-source comparator audit and documented adaptation choices.
• results/seven_method_benchmark/ - audited benchmark summary tables (no proprietary observations).
• tests/ - deterministic numerical, public-API, and benchmark source-audit tests.
• REAL_DATA_SCHEMA.md - construction of the 37 application variables.

Reproduce the main DAGGuard analyses

Fast checks and synthetic application twin:

bash reproduce_submission.sh

Full public simulation suite:

DAGGUARD_FULL=1 bash reproduce_submission.sh

Authorized real-data analysis:

python realdata_postselection_diagnostics.py \
  --data /authorized/path/train2023cw_simple.csv \
  --out results/swine_application \
  --primary-only

Seven-method benchmark provenance

The benchmark distinguishes the methods' inferential targets and implementation status. NOTEARS was checked against the public xunzheng/notears linear implementation. Li & Wang PC-FDR is an independent implementation of the published Algorithm 3. PC-p is a source-audited Python port of the authors' official MATLAB code because MATLAB/Octave was unavailable in the benchmark runtime. The Wang et al. (2026) structural method is a published-parameter adaptation to continuous variables by empirical tertiles and is not represented as official author software. PC-family procedures and the Wang adaptation are compared by skeleton adjacency because they need not return a uniquely oriented DAG comparable to NOTEARS/DAGGuard.

See benchmarks/seven_method/SOURCE_AUDIT.md, benchmarks/seven_method/README.md, results/seven_method_benchmark/method_implementation_provenance.csv, and results/seven_method_benchmark/AUDIT_NOTES.md.

Tests

python -m unittest discover -s tests -v
python -m examples.dagguard_quickstart
python synthetic_application_twin.py --out results/synthetic_application_twin

The public API tests cover scale invariance for exact refinement, greedy refinement, and pruning pressure; duplicate columns; near ties; saturated local models; constant responses; and exact or numerically near-exact fits. Additional regression tests check the Gaussian Fisher-z formula, PC-FDR step-up rule, PC-p BY estimator, and the discrete-BIC parameter count used in the Wang adaptation.

Data availability

The commercial swine observations are proprietary and are not included. Requests for access remain subject to the applicable data-use agreements. Only non-row-level derived summaries are public.

Historical reproducibility

This repository was renamed from NOTEARS-BP to DAGGuard on August 24, 2026. The application results used in the manuscript remain tied to historical commit 509cb29c24967d12b99e2b53641349ce7bb470ed; preserving that commit keeps the analysis audit trail intact. GitHub redirects the earlier repository URL to the current DAGGuard repository.

Citation

Wang M, Liu P, Magalhaes ES, Wang C. DAGGuard: Exact and Greedy BIC Refinement of Learned DAGs with an Application to Commercial Swine Production. Journal of Data Science, submitted.