# 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]( "Causal inference in statistics: An overview; Statist. Surv."). - **Introduction to causal inference** - Separates defining a causal question, stating assumptions, identifying an estimand, and estimating it; covers interventions, counterfactuals, and mediation. [Pearl 2010]( "An Introduction to Causal Inference; Int. J. Biostat."). - **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]( "Causality and Statistical Learning; Am. J. Sociol."). - **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]( "Causal discovery and inference: concepts and recent methodological advances; Appl Inform"). - **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]( "A Survey of Learning Causality with Data: Problems and Methods; ACM CSUR"). - **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]( "Bayesian Causal Inference: A Critical Review; Philosophical Transactions A"). - **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]( "Statistical Tools for Causal Inference; book"). ## 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]( "On the Application of Probability Theory to Agricultural Experiments. Essay on Principles. Section 9; Statist. Sci."). - **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]( "Causal Inference Using Potential Outcomes: Design, Modeling, Decisions; JASA"), [Rubin 1974]( "Estimating causal effects of treatments in randomized and nonrandomized studies.; J. Educ. Psychol."). - **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]( "The central role of the propensity score in observational studies for causal effects; Biometrika"). - **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 of Causal Effects Using Instrumental Variables; JASA"). - **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]( "Identification of Conditional Interventional Distributions; UAI"). - **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]( "Causal inference in statistics: An overview; Statist. Surv."). - **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]( "An Introduction to Proximal Causal Learning; arXiv"). ## 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]( "A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect; Mathematical Modelling"), [Robins et al. 2000]( "Marginal Structural Models and Causal Inference in Epidemiology; Epidemiology"). - **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]( "Recursive Partitioning for Heterogeneous Causal Effects; PNAS"). - **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]( "Estimation and Inference of Heterogeneous Treatment Effects using Random Forests; Journal of the American Statistical Association"). - **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]( "Double/Debiased Machine Learning for Treatment and Causal Parameters; The Econometrics Journal"). - **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]( "Approximate Residual Balancing: De-Biased Inference of Average Treatment Effects in High Dimensions; JRSS B"). - **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]( "Generalized Random Forests; The Annals of Statistics"). - **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]( "Meta-learners for Estimating Heterogeneous Treatment Effects using Machine Learning; PNAS"). - **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]( "Quasi-Oracle Estimation of Heterogeneous Treatment Effects; Biometrika"). - **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]( "Towards optimal doubly robust estimation of heterogeneous causal effects; Electronic Journal of Statistics"). - **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]( "Impossibility of Distribution-Free Predictive Inference for Individual Treatment Effects; NeurIPS"). ## 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]( "Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program; JASA"). - **Causal mediation** - Separates definition, identification, estimation, and sensitivity analysis for mediation effects, accommodating nonlinear models and making identifying assumptions explicit. [Imai et al. 2010]( "A general approach to causal mediation analysis.; Psychological Methods"). - **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]( "Making Sense of Sensitivity: Extending Omitted Variable Bias; JRSS B"). - **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]( "Difference-in-Differences with Multiple Time Periods; Journal of Econometrics"). ## 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]( "Causal Inference in the Presence of Latent Variables and Selection Bias; UAI"). - **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]( "Optimal Structure Identification With Greedy Search; JMLR"). - **LiNGAM** - Uses independent component analysis to identify linear acyclic causal structure under non-Gaussian disturbances and no hidden confounders. [Shimizu et al. 2006]( "A Linear Non-Gaussian Acyclic Model for Causal Discovery; JMLR"). - **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]( "Estimating High-Dimensional Directed Acyclic Graphs with the PC-Algorithm; JMLR"). - **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]( "Copula PC Algorithm for Causal Discovery from Mixed Data; ECML PKDD"). - **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]( "Causal discovery of linear acyclic models with arbitrary distributions; UAI"). - **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]( "Nonlinear causal discovery with additive noise models; NeurIPS"). - **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]( "DAGs with NO TEARS: Continuous Optimization for Structure Learning; NeurIPS"). ## 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]( "A General Algorithm for Deciding Transportability of Experimental Results; J. Causal Inference"). - **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]( "Causal inference using invariant prediction: identification and confidence intervals; JRSS B"). ## 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]( "Investigating Causal Relations by Econometric Models and Cross-spectral Methods; Econometrica"). - **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]( "Analyzing multiple nonlinear time series with extended Granger causality; Physics Letters A"). - **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]( "Search for Additive Nonlinear Time Series Causal Models; JMLR"). - **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]( "Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets; UAI"). - **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]( "Large-scale nonlinear Granger causality: A data-driven, multivariate approach to recovering directed networks from short time-series data; Scientific Reports"). ## 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]( "Towards Causal Representation Learning; arXiv"). ## 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](). ## Related topics - [Bayesian Networks](https://mlpapers.org/bayesian-nets/) - [Bayesian Inference](https://mlpapers.org/bayesian-inference/) - [Models](https://mlpapers.org/models/)