Causal Discovery beyond Conditional Independence: Identifiability, Algorithms, and Statistical Inference for Directed Acyclic and Cyclic Graphs

Yang Ni

stat.MEOct 16, 2026

Causal Discovery beyond Conditional Independence: Identifiability, Algorithms, and Statistical Inference for Directed Acyclic and Cyclic Graphs

Authors: Yang Ni

Organizations: Department of Statistics and Data Science, The University of Texas at Austin

Abstract

Learning causal structure from data requires assumptions about mechanisms, noises, observation schemes, and interventions. Historically, causal discovery has relied primarily on conditional independence. This review focuses on modern causal discovery methods that exploit properties of distributions beyond conditional independence. Our organizing principle separates the source of identification from the statistical estimator and its computational implementation. This perspective connects conditional-independence methods with distributional asymmetries, conditional-moment restrictions, and variation across environments or individuals. We examine discovery under latent confounding and feedback, distinguishing causal information about observed variables from reconstruction of latent structure and emphasizing the additional semantics required by cyclic models. We also distinguish population identification, frequentist and Bayesian consistency, uncertainty quantification, and algorithmic guarantees. Throughout, examples show why the same observational distribution can support different conclusions under different model classes. We conclude with guidance for comparing methods at their identifiable targets and a research agenda for uncertainty, heterogeneous populations, non-scalar variables, and the intersection of discrete data, feedback, and hidden confounding.

Figures & tables

Family and data	Hidden / cycles	Target and key restriction	Uncertainty to report
PC / GES; continuous or categorical with suitable test/score ( Kalisch and Bühlmann, 2007 ; Chickering, 2002 )	No / No	CPDAG; Markov and faithfulness or the respective score conditions	Unresolved orientations
LiNGAM; continuous ( Shimizu et al., 2006 )	No / No	DAG; linearity and independent non-Gaussian errors	Separate ordering-test UQ under its assumptions ( Wang et al., 2026 ) ; see Section 10.4
ANM / CAM; continuous ( Peters et al., 2014 ; Bühlmann et al., 2014 )	No / No	DAG in restricted additive-noise classes; excludes reversible mechanisms	Ordering confidence sets require additional fit/test conditions; selection alone supplies no coverage
Equal-variance Gaussian models ( Peters and Bühlmann, 2014 )	No / No	DAG; common error variance in the specified units	Effect test inversion: asymptotic or split-likelihood coverage ( Strieder and Drton, 2023 )
Count moments; Poisson / QVF ( Park and Raskutti, 2015 ; Park and Raskutti, 2018 )	No / No	Order and DAG; conditional family, dispersion restriction, and nondegeneracy	Selection consistency differs from calibrated graph or effect confidence
Categorical / mixed mechanisms ( Ni and Mallick, 2022 ; Ni, 2022 ; Choi and Ni, 2023 ; Zeng et al., 2022 )	No / No	Direction or DAG; ordinal, conditional-family, support, or constructed mixed-model restrictions	Bayesian fits permit posterior summaries

Table 1: Choosing a method by its statistical assumptions. “Hidden / cycles” describes the cited model, not an entire research area. Equivalence-class ambiguity, posterior uncertainty, and confidence coverage are different quantities. UQ entries name available interpretations or separate inferential extensions, not guarantees of every algorithm in a row.

Causal Discovery

Causal discovery infers cause-and-effect structures from observational and interventional data, distinguishing causal relationships from correlation.

See all 221 papers

Nov 13, 2023

Causal Discovery in Mixtures of Populations

Causal discovery aims to learn causal structures up to certain symmetries. Diverse populations or changing environments give rise to heterogeneous data in the following sense: each population/environment is a ``source'' which idiosyncratically determines the forms of causal effects. From this perspective, the source is a latent common cause for every observed variable. While some methods for causal discovery can work around latent confounding in special cases, a global confounder poses a significant challenge. The only known ways to deal with latent global confounding involve making assumptions that limit structural equations and/or noise functions. We demonstrate that globally confounded causal structures can still be identified with arbitrary structural equations and noise functions, so long as the number of latent classes remains small relative to the size and sparsity of the underlying DAG. The approach relies on agglomerating variables into large-enough matrices of moments, whose ranks directly reveal graphical properties of the causal structure. We also provide a statistical test to test the rank of these matrices.

Bijan Mazaheri, Spencer Gordon, Yuval Rabani +1

Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire, USA · Fragment Data Technologies, San Francisco, California, USA · Department of Computer Science, Hebrew Univeristy of Jerusalem, Jerusalem, Israel +1

Latent Variable Models Causal Discovery Causal Identifiability Causal Discovery with Latent Confounding Causal Inference with Unobserved Confounding

Ask PDF

Sep 28, 2026

The Statistical Cost of Causal Discovery with Feedback

What determines the unavoidable sample cost of learning cyclic causal structure? For cyclic linear non-Gaussian models, we study exact condensation recovery from observational data: identifying the strongly connected component (SCC) partition and all edges between components. We establish the first information-theoretic lower bounds on sample complexity for this target. For

p

variables, maximum SCC size

s_{\max}

, and maximum external-parent count

d_B

, any estimator requires order

s_{\max}\log(ep/s_{\max})+d_B\log(ep/d_B)

samples in the worst case over a regular model class. These bounds distinguish the costs of SCC membership and external-parent selection. Under principal invertibility and without correlation faithfulness, we establish a population block-exogeneity principle that identifies unknown root SCCs through residual independence and inclusion minimality. A sparse-adjustment characterization shows that small adjustment sets suffice to identify SCCs and their direct external parents, without regressing on all previously recovered variables. These characterizations yield BlockExo, which attains a structurally matching sample bound without knowing

s_{\max}

d_B

under suitable conditions. Simulations support the structural dependence of our sample bound and demonstrate BlockExo's sample-efficient recovery in comparisons with other methods for cyclic causal discovery.

Sunmin Oh, Seungsu Han, Gunwoong Park

Department of Statistics; Institute for Data Innovation in Science, Seoul National University, Korea · Department of Operations Research and Financial Engineering, Princeton University, USA · Department of Statistics; Interdisciplinary Program in Artificial Intelligence; Institute for Data Innovation in Science, Seoul National University, Korea.

Sample Complexity Causal Discovery Causal Structure Learning

Ask PDF

Mar 5, 2026

Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning

Causal discovery with latent variables is a fundamental task. Yet most existing methods rely on strong structural assumptions, such as enforcing specific indicator patterns for latents or restricting how they can interact with others. We argue that a core obstacle to a general, structural-assumption-free approach is the lack of an equivalence characterization: without knowing what can be identified, one generally cannot design methods for how to identify it. In this work, we aim to close this gap for linear non-Gaussian models. We establish the graphical criterion for when two graphs with arbitrary latent structure and cycles are distributionally equivalent, that is, they induce the same observed distribution set. Key to our approach is a new tool, edge rank constraints, which fills a missing piece in the toolbox for latent-variable causal discovery in even broader settings. We further provide a procedure to traverse the whole equivalence class and develop an algorithm to recover models from data up to such equivalence. To our knowledge, this is the first equivalence characterization with latent variables in any parametric setting without structural assumptions, and hence the first structural-assumption-free discovery method. Code and an interactive demo are available at https://equiv.cc.

Haoyue Dai, Immanuel Albrecht, Peter Spirtes +1

Carnegie Mellon University · FernUniversität in Hagen · MBZUAI

Causal Discovery Causal Discovery with Latent Confounding Causal Identifiability Structural Causal Models

Ask PDF

Abstract

Causal Discovery

Explore similar work

Causal Discovery in Mixtures of Populations

The Statistical Cost of Causal Discovery with Feedback

Distributional Equivalence in Linear Non-Gaussian Latent-Variable Cyclic Causal Models: Characterization and Learning