Differentiable Causal Discovery of Linear Non-Gaussian Acyclic Models Under Unmeasured Confounding

TMLR Paper4056 Authors

Abstract: We propose a score-based method that extends the framework of the linear non- Gaussian acyclic model (LiNGAM) to address the problem of causal structure estimation in the presence of unmeasured variables. Building on the method pro- posed by Bhattacharya et al. (2021), we develop a method called ABIC LiNGAM, which assumes that error terms follow a multivariate generalized normal distribu- tion and employs continuous optimization techniques to recover acyclic directed mixed graphs (ADMGs). We demonstrate that the proposed method can esti- mate causal structures, including the possibility of identifying their orientations, rather than only Markov equivalence classes, under the assumption that the data are linear and follow a multivariate generalized normal distribution. Additionally, we provide proofs of the identifiability of the parameters in ADMGs when the er- ror terms follow a multivariate generalized normal distribution. The effectiveness of the proposed method is validated through simulations and experiments using real-world data.

Submission Length: Long submission (more than 12 pages of main content)

Changes Since Last Submission:

Based on the comments we received, we have made the following revisions. The differences from the initial submission are color-coded so that the changes are clear.

Reviewer 1
①“ABIC LiNGAM vs. BANG: tiny performance gap”
Even a small empirical gap masks four practical advantages of ABIC LiNGAM:

Single score optimisation: avoids multiple statistical tests and type-I/II error propagation.
Easy injection of prior knowledge: domain constraints can be added as simple numerical penalties.
Unified Gaussian / non-Gaussian treatment: one framework, Markov-equivalence for Gaussian, full orientation for non-Gaussian.
High extensibility: non-linearities or mixed data types require only loss-function tweaks, not new tests.

②Background vs. novel contribution unclear
Move the description of ABIC (Bhattacharya et al., 2021) to the Related Work section.
Keep §4.2 focused exclusively on our new extensions (non-Gaussian errors, generalised-normal parameters).

③Behaviour under Gaussian errors
With Gaussian errors the method recovers one member of the Markov equivalence class (directions unidentifiable), but the same optimisation routine remains valid, preserving a coherent workflow.

④Non-linear structural functions
Extra experiments with polynomial terms show high skeleton F-score (0.901) but lower edge-direction accuracy; BANG sometimes outperforms here. Future work: non-linear loss terms and theoretical guarantees.

⑤Linear structure + heavy-tailed errors
When the true shape parameter β is known (e.g. t-distribution, ν = 3) ABIC LiNGAM is best on all metrics. If β is estimated and mis-specified, direction accuracy drops—practitioners should grid-search β or use prior information.

Reviewer 2
①Literature coverage too narrow
Added ADMG work (Richardson & Spirtes 2002; Tashiro et al. 2014) and recent score-based methods (Nowzohour 2017; Bernstein 2020; Chen 2021; Claassen 2022; Ng 2024).
Contrasted our method with BANG and ABIC, stressing that non-Gaussianity enables full identification.

②Redundant text / missing definitions
Removed duplicated Brito & Pearl (2002) sentences; created a glossary subsection for “bow-free ADMG”, “Markov equivalence”, “non-Gaussianity”.

③Algorithm 2 resembles prior work
ABIC details now live in Related Work; Algorithm 2 explicitly highlights the new β≠1 term (handles non-Gaussian cases).

④⑤Notation issues
Corrected 　Z_{-i} = Ω_{-i, -i}^{-1}𝜀_{-1} defined tr(A) right after Eq. (22).

⑥Algorithm formatting
Split into stages (A) pseudo-variables, (B) parameter optimisation, (C) convergence; indented and annotated differences between Algorithms 1 & 2.

⑦Lack of structure / emphasis
Inserted Key Point and (Intuition) callouts at critical junctures; bolded main take-aways and used bullet lists.

⑧Large-sample anomalies (Fig. 3)
Explained that bigger samples increase power and over-detect edges when the penalty is fixed; appendix now shows 2k & 5k sample results and how stricter penalties restore precision.

⑨Need a consistency theorem
Stated plan to prove skeleton consistency for bow-free ADMGs with non-Gaussian errors in future work; current manuscript provides empirical evidence only.

Reviewer 3
Added two explicit statements (end of Introduction and end of Conclusion) underscoring that introducing non-Gaussian errors is a deliberate strategy, not a mere relaxation: higher-order moments unlock causal directions even with latent confounding.

Overall:
(i) foreground the methodological advantages and extensibility of ABIC LiNGAM
(ii) restructure the manuscript to cleanly separate prior work from contributions
(iii) fix notation/formatting
(iv) expand experiments and practical guidance
(v) emphasise why non-Gaussianity is pivotal for full identifiability.

Assigned Action Editor: ~Sergey_Plis1

Submission Number: 4056