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Full Causal Graph Recovery Under Arbitrary Nonlinear Latent Confounding

Mathematical Statistics Probability Theory Learning Theory

Sourced from the work of Daniele Tramontano, Jalal Etesami, Mathias Drton

§ Problem Statement

Setup

Let $X=(X_1,\dots,X_p)^\top$ be generated by an acyclic linear SEM on observed nodes $V=\{1,\dots,p\}$ ,

X_i=\sum_{j\in\operatorname{pa}_G(i)} b_{ij}X_j+N_i,

with DAG $G$ and non-Gaussian disturbances that may be jointly dependent through latent nonlinear mechanisms.

This setup follows Tramontano et al. (2025).

Source-supported facts: the cited paper establishes a necessary-and-sufficient graphical criterion (with a polynomial-time algorithm) for generic identifiability of direct causal effects in the acyclic setting, and reports additional results on causal-graph identifiability.

Unsolved Problem

For each DAG $G$ , write parameters as $(b,\eta)\in\Theta_G=\mathcal B_G\times\mathcal H_G$ , where $\mathcal B_G\subseteq\mathbb R^{|E(G)|}$ are directed-edge coefficients and $\eta$ collects nuisance latent/noise parameters. For fixed $\eta$ , define

\mathcal A_{G,\eta}:=\left\{b\in\mathcal B_G:\exists\,G'\not\sim G,\exists\,(b',\eta')\in\Theta_{G'}\text{ with }P_{G,b,\eta}=P_{G',b',\eta'}\right\}.

Call $G$ coefficient-generically identifiable if $\lambda_{|E(G)|}(\mathcal A_{G,\eta})=0$ for every admissible $\eta$ (equivalently, for almost every $b$ given $\eta$ ). Open question: characterize which observed-node DAGs are coefficient-generically identifiable in this arbitrary nonlinear latent-confounding model, and when only a coarser graph-equivalence class is identifiable.

§ Discussion

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§ Significance & Implications

Recovering the full DAG (or its sharp observationally identifiable equivalence class) is strictly stronger than identifying individual direct effects. Clarifying this boundary would extend the paper's direct-effect identifiability theory to end-to-end structure learning under general latent confounding.

§ Known Partial Results

Tramontano et al. (2024): Source-supported: in the acyclic setting, the paper proves a necessary-and-sufficient graphical criterion (with polynomial-time certification) for generic identifiability of direct effects, and the abstract reports additional causal-graph identifiability results. A full DAG-level necessary-and-sufficient characterization under arbitrary nonlinear latent confounding is not explicitly stated in that cited source sentence and is treated here as open.

§ References

[1]

Parameter identification in linear non-Gaussian causal models under general confounding

Daniele Tramontano, Jalal Etesami, Mathias Drton (2024)

arXiv preprint

📍 Abstract, p. 1 (arXiv:2405.20856v1), final sentence stating the paper "provide[s] new results on the identifiability of the causal graph."

Canonical citation target uses the base arXiv identifier (no version suffix) for metadata stability.

Link ↗arXiv ↗

§ Tags

causal-discovery graph-identifiability dag non-gaussianity latent-variables

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