Causal Inference

Frameworks and surveys

  • Causal inference in statistics - Connects structural models, interventions, counterfactuals, and potential outcomes; emphasizes that causal conclusions depend on assumptions beyond observed associations. Pearl 2009.
  • Introduction to causal inference - Separates defining a causal question, stating assumptions, identifying an estimand, and estimating it; covers interventions, counterfactuals, and mediation. Pearl 2010.
  • Causality and statistical learning - Gelman’s review distinguishes estimating effects of specified interventions from explaining observed outcomes, comparing graphical and potential-outcome approaches. Gelman 2011.
  • Causal discovery and inference survey - Reviews conditional-independence and structural-equation approaches, their identifiability assumptions, and extensions from independent observations to time series. Spirtes and Zhang 2016.
  • Learning causality with data - Surveys treatment-effect estimation and causal-relation discovery, examining where larger datasets help and where identification still requires substantive assumptions. Guo et al. 2020.
  • Bayesian causal inference - Reviews potential-outcome estimands, priors, identification, overlap, and sensitivity analysis; also considers instrumental variables and time-varying treatments. Li et al. 2023.
  • Statistical Tools for Causal Inference - Sylvain Chabé-Ferret’s open text covers causal estimands, identification, estimators, uncertainty, and research design. The linked edition is dated 2025. Chabé-Ferret 2025.

Potential outcomes and identification

  • Neyman’s field experiments - English translation of the 1923 work: represents potential yields by crop variety and plot and derives randomization-based uncertainty, although only one potential yield per plot is observed. Splawa-Neyman et al. 1990.
  • Potential outcomes: design and modeling - Defines effects through comparisons of potential outcomes and models treatment assignment separately; connects experimental design, Bayesian modeling, and decisions based on causal estimates. Rubin 2005, Rubin 1974.
  • Propensity scores - The treatment probability given observed covariates is a balancing score. Causal adjustment requires strongly ignorable assignment, including overlap; balancing observed covariates does not remove hidden confounding. Rosenbaum and Rubin 1983.
  • Instrumental variables - Under the paper’s identification assumptions, the instrumental-variable estimand identifies an average effect for compliers. Without those assumptions, the ratio lacks that causal interpretation. Angrist et al. 1996.
  • Identification beyond adjustment - Shpitser and Pearl give a complete algorithm for identifying conditional intervention distributions from a causal graph and observational law. Some effects need more than adjustment; the algorithm assumes the graph, not its discovery. Shpitser and Pearl 2006.
  • Back-door adjustment - Selects covariates that block back-door paths without including descendants of treatment. The criterion depends on a causal graph; adding more adjustment variables is not automatically helpful. Pearl 2009.
  • Proximal causal learning - Identifies effects using proxies for hidden confounders under proxy-independence, confounding-bridge and completeness conditions; sufficiently informative proxies replace full confounder measurement. Tchetgen et al. 2020.

Treatment-effect estimation with machine learning

  • Longitudinal causal effects - The g-formula identifies effects of treatment histories. MSMs model these effects and can be fitted with inverse-probability weights. Causal identification requires consistency, treatment support and no unmeasured confounding. Robins 1986, Robins et al. 2000.
  • Causal trees - Partitions covariate space by treatment-effect heterogeneity. Honest estimation uses separate samples to choose partitions and estimate leaf effects, under randomized assignment or unconfoundedness. Athey and Imbens 2016.
  • Causal forests - Extends random forests to heterogeneous treatment effects with honest trees; consistency and uncertainty results require unconfoundedness and the paper’s regularity assumptions. Wager and Athey 2018.
  • Double/debiased machine learning - Combines orthogonal scores with cross-fitting to reduce sensitivity to nuisance-model errors. Valid causal interpretation still requires identification assumptions; predictive accuracy alone is insufficient. Chernozhukov et al. 2018.
  • Approximate residual balancing - Combines balancing weights and regularized outcome regression for average treatment effects in high-dimensional linear models, with overlap and unconfoundedness assumptions. Athey et al. 2018.
  • Generalized random forests - Learns adaptive neighborhood weights for local moment equations, supporting heterogeneous treatment effects, instrumental-variable effects, and other targets beyond ordinary outcome prediction. Athey et al. 2019.
  • Meta-learners and the X-learner - Wraps supervised learners to estimate conditional average treatment effects. The X-learner exploits structure and unequal treatment-group sizes; no compared learner is uniformly best. Künzel et al. 2019.
  • R-learner - First estimates outcome and treatment nuisance functions, then optimizes a residualized treatment-effect objective; its quasi-oracle result is proved for a specified penalized-kernel setting. Nie and Wager 2020.
  • Doubly robust CATE estimation - Analyzes two-stage treatment-effect estimators and local polynomial double-residual regression; gives error bounds and conditions for oracle efficiency rather than assuming flexible learners guarantee it. Kennedy 2023.
  • Distribution-free ITE limits - Under ignorability/overlap with a continuous covariate and unrestricted continuous outcomes, uniformly valid individual-effect sets have infinite expected length. Procedures using a known randomization design are outside this result. Tao et al. 2026.

Policy evaluation and mechanisms

  • Synthetic controls - Constructs a weighted comparison from untreated units to match a treated unit before intervention; comparative-case inference depends on a credible donor pool and pre-intervention fit. Abadie et al. 2010.
  • Causal mediation - Separates definition, identification, estimation, and sensitivity analysis for mediation effects, accommodating nonlinear models and making identifying assumptions explicit. Imai et al. 2010.
  • Sensitivity to omitted confounding - Extends omitted-variable-bias analysis for regression. The robustness value quantifies how strongly unobserved confounding would need to relate to treatment and outcome to change a conclusion. Cinelli and Hazlett 2019.
  • Difference-in-differences with staggered timing - Identifies group/time effects under conditional parallel trends and supports explicit aggregation across treatment cohorts, avoiding a single effect summary that hides timing heterogeneity. Callaway and Sant’Anna 2021.

Graph discovery from independent observations

  • FCI and latent variables - Studies causal conclusions from conditional independences when latent variables and selection bias may exist; identifies features supported by the assumptions rather than promising a unique complete causal graph. Spirtes et al. 1995.
  • Greedy equivalence search - Searches over DAG equivalence classes with locally scored operators. Large-sample recovery guarantees depend on distributional and scoring conditions, not merely finding a high-scoring graph. Chickering 2002.
  • LiNGAM - Uses independent component analysis to identify linear acyclic causal structure under non-Gaussian disturbances and no hidden confounders. Shimizu et al. 2006.
  • PC algorithm - Uses conditional-independence information to estimate a graph skeleton and equivalence class. High-dimensional consistency is established for sparse Gaussian DAGs under stated assumptions. Kalisch and Bühlmann 2007.
    • Copula PC - Extends PC to mixed continuous/discrete data under a Gaussian-copula model, estimating correlations through rank-based Gibbs sampling before graph discovery. Cui et al. 2016.
  • Partially Gaussian linear models - Combines conditional-independence and independent-component approaches, characterizing which linear acyclic structures remain distinguishable when some disturbances are Gaussian. Hoyer et al. 2008.
  • Nonlinear additive-noise models - Uses restrictions from additive independent noise to infer causal direction in nonlinear systems; identifiability depends on the functional and noise assumptions. Hoyer et al. 2008.
  • NOTEARS - Expresses DAG acyclicity as a smooth equality constraint for continuous structure optimization. An acyclic fitted graph alone does not establish that its arrows are causal. Zheng et al. 2018.

Discovery across environments

  • Transportability - Uses graphical assumptions about population differences to combine source experiments with target observations. Bareinboim and Pearl decide when a causal effect can transfer; predictive domain adaptation need not identify that effect. Bareinboim and Pearl 2013.
  • Invariant causal prediction - Tests which predictor sets retain an invariant conditional distribution for a target across environments. Causal-parent confidence statements require the paper’s structural and intervention assumptions. Peters et al. 2016.

Time series and temporal discovery

  • Granger’s original formulation - Defines causality through improved prediction from past information. Apparent instantaneous effects can reflect coarse sampling or omitted variables; the result depends on the available information set. Granger 1969.
  • Extended Granger causality - Extends Granger’s predictive-dependence approach to nonlinear multivariate time series and evaluates it on chaotic systems; inferred direction remains tied to the observed variables and model. Chen et al. 2004.
  • Additive nonlinear time-series models - Combines conditional-independence tests from additive regressions with temporal ordering to recover causal information for a specified class of stationary nonlinear systems. Chu and Glymour 2008.
  • PCMCI+ - Selects conditioning sets for lagged and contemporaneous discovery in autocorrelated time series; assumes causal sufficiency, so latent-confounder settings require a different method. Runge 2020.
  • Large-scale nonlinear Granger causality - Uses nonlinear state-space transformations to estimate multivariate directed predictive interactions from short, high-dimensional series; evaluates simulated networks and fMRI recordings. Wismüller et al. 2021.

Causal representations

  • Causal representation learning - Frames discovery of causal variables from low-level observations as a distinct problem from learning a graph over given variables, linking causal assumptions to transfer and generalization. Schölkopf et al. 2021.

Software

  • Tigramite - Time-series causal discovery and effect estimation with linear and nonparametric conditional-independence tests; includes distinct methods for different temporal and confounding assumptions. docs.
  • DoWhy - Separates causal modeling, identification, and estimation, with sensitivity and refutation tools; also supports structural causal models, interventions, and counterfactual queries. docs.
  • EconML - Implements heterogeneous-effect estimators, including double machine learning, forests, meta-learners, and instrumental-variable methods, with inference and validation interfaces. docs.