Recovering Markov boundary -- the minimal set of variables that maximizes predictive performance for a response variable -- is crucial in many applications. While recent advances improve upon traditional constraint-based techniques by scoring local causal structures, they still rely on nonparametric estimators and heuristic searches, lacking theoretical guarantees for reliability. This paper investigates a framework for efficient Markov boundary discovery by integrating conditional entropy from information theory as a scoring criterion. We design a novel masked autoregressive network to capture complex dependencies. A parallelizable greedy search strategy in polynomial time is proposed, supported by analytical evidence. We also discuss how initializing a graph with learned Markov boundaries accelerates the convergence of causal discovery. Comprehensive evaluations on real-world and synthetic datasets demonstrate the scalability and superior performance of our method in both Markov boundary discovery and causal discovery tasks.
Figures & tables
Fig. 1: Illustration of weight masking mechanism in FANS , as described in Subsection III-B2 , for specific subset S={X2,X3,X5} . This simplifies the original illustration of MADE architecture for better visualization, showing only a hidden layer with a block of nodes and output layer with a block of nodes. Full autoencoder may consist of multiple hidden layers, each with several blocks of hidden nodes, which can be permuted, as shown in Fig. 1 from [ 15 ] . Solid connections represent fully-connected weight regions in Step 1, while red connections indicate autoregressive masks in Step 2. Other subsets will have different masks. In compact version (right figure), assuming maximum size of observed subsets is 3, each block of hidden layer reduces to 2 nodes. Original element indices of S are rescaled and colliders resolved: [2,3,5]→[⌈5−12∗2⌉,⌈5−13∗2⌉,⌈5−15∗2⌉]→[1,2,3] .
Dataset
Nodes
Edges
Observations
Sachs [ 34 ]
11
17
853
SynTReN [ 40 ]
20
24
500
SERGIO [ 10 ]
100
137
9000
ARTH150 [ 28 ]
107
150
5000
TABLE I: Real datasets
Graph type
Metric
IAMB [ 38 ]
S2TMB [ 14 ]
EEMB [ 42 ]
TC [ 32 ]
KMB [ 43 ]
DAT-Moral [ 2 ]
FANSITEMB (ours)
Linear data
d30
nDCG ( ↑ )
47.18 ± 12.81
93.01 ± 3.74
90.78 ± 6.54
98.91 ± 1.25
70.58 ± 7.60
90.00 ± 3.84
99.34 ± 1.02
AveP ( ↑ )
40.34 ± 15.41
90.56 ± 5.11
87.42 ± 8.93
98.42 ± 1.81
60.69 ± 9.57
86.14 ± 5.17
99.08 ± 1.39
F1 ( ↑ )
48.25 ± 12.72
93.58 ± 3.79
91.39 ± 6.40
98.68 ± 1.72
65.36 ± 7.04
21.28 ± 3.22
99.44 ± 0.87
d100-1
nDCG ( ↑ )
41.71 ± 5.36
94.27 ± 1.90
96.01 ± 2.00
99.90 ± 0.22
−
97.42 ± 1.34
99.86 ± 0.22
AveP ( ↑ )
34.54 ± 5.37
91.80 ± 2.74
94.29 ± 2.88
99.85 ± 0.34
−
96.33 ± 1.72
99.78 ± 0.33
TABLE II: Markov boundary discovery performance in % . The numbers are mean ± standard deviation over 5 independent simulations, except for real data. Missing entries ( − ) correspond to instances with runtime exceeding the limit of 10 hours. Bold : best performance, italic : second-best performance.
Graph type
Metric
DAGMA [ 3 ]
COSMO [ 24 ]
SDCD [ 26 ]
DAT [ 2 ]
MARVEL [ 25 ]
FANSITE-DCD (ours)
Nonlinear data
d30-G
SHD ( ↓ )
15.4 ± 8.85
25 ± 5.66
13 ± 5.43
420.2 ± 7.50
24.6 ± 6.23
12.8 ± 6.53
AUC-ROC ( ↑ )
0.752 ± 0.095
0.822 ± 0.084
0.858 ± 0.074
0.502 ± 0.068
0.603 ± 0.039
0.902 ± 0.045
AUC-PR ( ↑ )
0.488 ± 0.172
0.375 ± 0.077
0.570 ± 0.144
0.036 ± 0.013
0.144 ± 0.061
0.592 ± 0.098
d30-MN
SHD ( ↓ )
21.8 ± 7.82
48.4 ± 12.46
30.6 ± 9.81
419.8 ± 6.10
26.6 ± 8.91
18.2 ± 8.14
AUC-ROC ( ↑ )
0.654 ± 0.066
0.777 ± 0.058
0.766 ± 0.037
0.488 ± 0.061
0.578 ± 0.041
0.866 ± 0.044
TABLE III: Causal discovery performance . The numbers are mean ± standard deviation over 5 independent simulations, except for real data. Bold : best performance, italic : second-best performance.
Fig. 2: Different model options for estimating our score criterion—conditional entropy. Experiments conducted on synthetic nonlinear graphs with d=100 . FANS-CPT, FANS-FN, and FANS-FN-NUL represent three variants in our proposed class of masked autoregressive flows (MAFs) tailored for MB discovery, corresponding to: a compact version with leaf masks, a non-compact version with leaf-masks, and a non-compact version without leaf masks. NAF and BNAF are typical MAF classes, while ACE [ 37 ] represents a class of energy-based estimators. The left figure illustrates the reduction across training epochs of ∑H(X∣MBX) , which is the total conditional entropy as each variable conditions on its MB)—smaller values indicate better performance. The right figure shows the number of trainable parameters; fewer parameters imply greater efficiency. When FANS is fully parameterized and trained with leaf-masking (FANS-FN), it learns even better than the compact counterpart. However, the compact FANS , with half the parameters of NAF, is expressive enough to surpass NAF in training and estimation, demonstrating the efficacy of our FANS class, as discussed in Subsection III-B2 .
Leveraging deep learning for causal discovery in time series remains challenging because existing neural methods predominantly rely on component-wise architectures that fail to capture shared system dynamics or employ decoupled post-hoc graph extraction that risks overfitting to spurious correlations. We propose Mask2Cause, an end-to-end framework that recovers the underlying causal graph directly during the forecasting forward pass. Our approach introduces an Inverted Variable Embedding and an Adjacency-Constrained Masked Attention mechanism, trained with homoscedastic or heteroscedastic objectives to capture causal influences in both mean and variance. Empirical results on diverse benchmarks, from synthetic chaotic dynamics to realistic biological simulations, demonstrate state-of-the-art causal discovery with significantly reduced parameter complexity compared to standard baselines. We further show that inferred causal structures can be used to reduce parameter count of forecasting models by more than 70% on average while maintaining predictive accuracy.
Omar Muhammad, Pasupuleti Dhruv Shivkant, Deepak N. Subramani
Causal discovery from observational data is a fundamental yet challenging task in scientific research. While existing approaches are primarily based on conditional independence tests, structure scores, or restrictive functional assumptions, we propose Decoupled Causal Discovery (DCD), a novel decoupling-based perspective that does not rely on these methodologies. DCD directly identifies the Markov boundary (MB) by decoupling non-target variables via weighting functions, such that only variables within the MB preserve dependence with the target under the decoupled distribution. Building on this, DCD iteratively constructs the Completed Partially Directed Acyclic Graph (CPDAG) by exploiting structural asymmetries within the MBs. We establish the theoretical identifiability, soundness, and completeness of DCD. Empirical evaluations demonstrate that DCD achieves strong performance, particularly excelling in challenging noise regimes.
Zhengkang Guan, Fei Wu, Kun Kuang
College of Computer Science and Technology Zhejiang University
Identifying causal bounds from discrete observational data is a fundamental problem in causal inference. While exact analytical bounds exist for certain graphs, this is not generally the case. In this paper, we introduce a framework for estimating such bounds using neural generative models together with a new extrapolation procedure to estimate these bounds more accurately. Across multiple causal inference settings, it produces bounds that are consistently closer to the true bounds than those obtained with existing general neural approaches. We validate the framework on a range of benchmark problems for which exact bounds are available.
Max Zhu, Srivathsan Amruth, Sina Akbari +1
Department of Pure Mathematics and Mathematical Statistics University of Cambridge