Causal Directed Acyclic Graph (DAG)

Each directed edge (arrow) represents a direct causal effect from one variable to another. The direction X → Y encodes the assumption that manipulating X will change Y, but not vice versa. This is fundamentally different from associational graphs, where edges merely indicate statistical dependence. The arrow's directionality is the primary mechanism for encoding asymmetric causal knowledge and is the basis for applying interventions via the do-operator.

A valid DAG contains no feedback loops or directed cycles. A path X → Y → Z → X is strictly forbidden. This constraint ensures that a variable cannot be a cause of itself, either directly or through a chain of intermediaries. Acyclicity is a prerequisite for:

• Defining a valid joint probability distribution via the causal Markov condition
• Applying d-separation criteria for testing conditional independencies
• Ensuring that recursive factorization of the distribution is well-defined

Each node in a DAG represents a random variable in the system under study. These can be:

• Observed variables: Measured data such as gene expression levels, blood pressure, or treatment status
• Unobserved (latent) variables: Confounders or mediators not present in the dataset, often represented with dashed circles or omitted entirely
• Deterministic nodes: Variables that are a fixed mathematical function of their parents The choice of which variables to include is the most critical modeling decision, as omitted variable bias arises directly from missing nodes.

A path is any sequence of edges connecting two nodes, regardless of arrow direction. Distinguishing path types is critical:

• Causal paths: All arrows point forward from cause to effect (e.g., X → M → Y). These transmit the causal effect of interest.
• Backdoor paths: Contain an arrow pointing into the exposure (e.g., X ← C → Y). These create confounding and must be blocked by adjustment.
• Collider paths: Contain two arrows pointing into a single node (e.g., X → S ← Y). These are naturally blocked but can open with improper conditioning, causing collider bias.

d-separation (directional separation) is the graphical criterion for reading conditional independencies from a DAG. Two sets of nodes are d-separated if every path between them is blocked. A path is blocked when:

• It contains a chain (X → M → Y) or fork (X ← C → Y) where the middle node is conditioned on
• It contains a collider (X → S ← Y) where neither the collider nor its descendants are conditioned on This property allows researchers to derive testable implications of their causal model from observational data.

The causal Markov condition states that a variable is independent of all its non-descendants given its direct causes (parents). This allows the joint probability distribution over all nodes to be factorized as: P(X₁, X₂, ..., Xₙ) = ∏ P(Xᵢ | parents(Xᵢ)) This modular decomposition is what makes DAGs computationally tractable. It implies that the causal system can be decomposed into autonomous mechanisms, each representing a stable physical process that remains invariant under interventions on other mechanisms.

A formal mathematical framework developed by Judea Pearl for reasoning about interventions. It uses three rules to transform expressions containing the do-operator into estimable quantities from observational data.

• Enables derivation of causal effects when randomized experiments are impossible
• Connects graphical structures (d-separation) to algebraic derivations
• Foundational for algorithms like ID and IDC that automate identification

A graphical criterion for reading conditional independencies directly from a DAG. A path between two nodes is blocked if it contains a chain or fork where the middle node is conditioned on, or a collider where neither the collider nor its descendants are conditioned on.

• Determines which variables must be adjusted for to isolate causal effects
• Underpins the back-door criterion for covariate selection
• If d-separation holds in the graph, statistical independence holds in the data (assuming faithfulness)

A graphical rule for identifying a sufficient set of covariates to adjust for when estimating the causal effect of an exposure on an outcome. A set of variables Z satisfies the back-door criterion if it blocks all back-door paths—paths that start with an arrow pointing into the exposure—and contains no descendants of the exposure.

• Provides a complete, non-parametric identification strategy
• Fails when unobserved confounders exist on back-door paths
• Contrast with the front-door criterion, which uses mediators when confounders are unmeasured

A systematic distortion that arises when conditioning on a common effect (a collider) of two variables. This conditioning induces a spurious association between the variables, even if they are completely independent in the population.

• Classic example: conditioning on hospitalization (a collider of disease and socioeconomic status) creates a false disease-poverty link in hospital data
• Known as Berkson's paradox in epidemiology and selection bias in econometrics
• A core threat to validity in observational studies and a key motivation for DAG-based study design

A class of algorithms that infers causal structures directly from observational data without requiring a pre-specified hypothesis. These algorithms test conditional independencies to constrain the space of possible DAGs.

• Constraint-based methods (e.g., PC Algorithm) use statistical tests to prune edges
• Score-based methods (e.g., GES) search for graphs that optimize a fit score
• Output is typically a Markov equivalence class—a set of DAGs that encode the same independencies and cannot be distinguished without further assumptions

A causal inference framework that estimates what would have happened to an individual's outcome if they had received a different exposure than the one they actually experienced. In the DAG framework, this requires a fully specified Structural Causal Model (SCM) with functional relationships, not just a qualitative graph.

• Goes beyond average treatment effects to answer individual-level "what if" questions
• Requires the three-step process: abduction, action, and prediction
• Essential for mediation analysis, fairness evaluation, and algorithmic recourse