Causal Graph

Nodes represent the measurable variables in the system under study. Each node corresponds to a distinct feature, attribute, or event.

• Types of Variables: Treatment (X), outcome (Y), confounders (Z), mediators (M), and instruments (I).
• Latent Variables: Often represented as unobserved nodes, indicating hidden common causes that create confounding.
• Example: In a graph modeling drug efficacy, nodes could be 'Drug Dosage', 'Blood Pressure', 'Age', and 'Recovery Status'.

Directed edges (arrows) represent assumed direct causal relationships. An edge from node A to node B signifies that A is a direct cause of B.

• Acyclicity: The graph must be acyclic—no path can start and end at the same node. This prevents causal loops and ensures well-defined probability distributions.
• Direct vs. Indirect Cause: A direct cause has an edge pointing to the effect. An indirect cause influences the effect through a chain of direct causes (a path).
• Missing Edge: The absence of a direct arrow is a strong assumption of no direct causal effect.

A path is any sequence of connected edges between nodes, regardless of direction. Paths are critical for understanding how association flows through the graph.

• Causal Path (Directed Path): A path where all edges point in the same direction (e.g., X → M → Y). This represents a mechanism of causal influence.
• Backdoor Path: A non-causal path connecting a treatment and outcome that remains open if not blocked by conditioning. It often creates spurious association via a common cause (e.g., X ← Z → Y).
• Frontdoor Path: A specific causal path that is mediated by an observed variable, useful for identification under unmeasured confounding.

d-separation ('d' for directional) is the fundamental graphical rule for reading conditional independencies from a causal graph. Two sets of nodes X and Y are d-separated by a third set Z if all paths between them are blocked by Z.

• A path is blocked by Z if it contains a chain (A → Z → B) or a fork (A ← Z → B) where the middle node Z is in the conditioning set.
• It is also blocked if it contains a collider (A → C ← B) where C and none of its descendants are in the conditioning set.
• Under the Causal Markov Condition, d-separation implies statistical independence in the observed data.

A collider is a node where two or more directed edges converge (e.g., A → C ← B). Colliders have unique, often counter-intuitive, statistical properties.

• Conditioning on a collider opens a path between its parents (A and B), inducing a statistical association even if they are independent. This is collider bias or Berkson's paradox.
• Example: In a graph 'Talent → Fame ← Luck', Talent and Luck are independent. But if we condition on Fame (study only famous people), a negative correlation between Talent and Luck may appear.
• Correctly identifying colliders is essential for causal discovery algorithms like the PC algorithm.

These terms describe the relative positions of nodes within the directed graph structure.

• Parents: The direct causes of a node (nodes with edges pointing into it).
• Children: The direct effects of a node (nodes it points to).
• Ancestors: All nodes that can reach a given node via a directed path (its parents, grandparents, etc.).
• Descendants: All nodes reachable from a given node via a directed path (its children, grandchildren, etc.).
• Non-Descendants: All nodes except the node itself and its descendants. Under the Causal Markov Condition, a node is independent of its non-descendants given its parents.

A Structural Causal Model (SCM) is the formal mathematical framework underpinning a causal graph. It consists of:

• A set of structural equations, one for each variable, defining how it is generated from its direct causes and an independent noise term.
• A corresponding causal graph (DAG) visualizing these dependencies.
• An explicit representation of interventions via the do-operator, allowing the model to answer "what if" questions by modifying equations.

While the graph shows the structure, the SCM provides the full functional and probabilistic specification needed for counterfactual reasoning.

Causal inference is the overarching goal: drawing conclusions about cause-and-effect relationships from data. The causal graph serves as the roadmap for this process. Key steps include:

• Identification: Using the graph (e.g., via backdoor or frontdoor criteria) to determine if a causal effect can be estimated from available data.
• Estimation: Applying statistical methods (matching, weighting, regression) to compute the effect.
• Unlike pure prediction, causal inference requires explicit assumptions about the data-generating process, encoded in the graph, to move beyond correlation.

An intervention, denoted do(X=x), is the act of forcibly setting a variable to a value, simulating an experiment. It modifies the causal graph by removing incoming edges to the intervened variable.

Do-calculus is a set of three inference rules that allow researchers to compute the probability of post-intervention outcomes (P(Y | do(X))) from observational data probabilities (P(Y | X)), provided the causal graph is known. It is the mathematical engine for translating "seeing" data into knowledge about "doing" actions.

Causal discovery refers to algorithms that attempt to learn the causal graph itself from data. Methods include:

• Constraint-based (e.g., PC algorithm): Tests for conditional independencies to infer graph structure.
• Score-based: Searches over graph space to optimize a fit score (e.g., BIC).
• Functional causal models: Assumes specific functional forms (e.g., linear non-Gaussian) to identify directionality.

These algorithms output a full or partial DAG, providing the hypothesized structure needed for subsequent causal inference. Results are always subject to assumptions (e.g., causal sufficiency, faithfulness).

These are graphical criteria for determining causal identifiability—whether an effect can be estimated from data.

• Backdoor Criterion: Identifies a set of variables Z that, when conditioned on, blocks all backdoor paths (non-causal, confounding paths) between treatment X and outcome Y. If such a set exists, the causal effect is identifiable via adjustment: P(Y | do(X)) = Σ_z P(Y | X, Z=z) P(Z=z).

• Frontdoor Criterion: Provides an identification formula when unmeasured confounding exists, by finding a mediator M that fully intercepts X's effect on Y and is unconfounded.

A counterfactual query represents the highest level of causal reasoning: "What would have happened if...?" It compares an observed outcome to the outcome in a hypothetical world where a cause was different.

• Formally, for an individual who received treatment X=1 and had outcome Y=1, the counterfactual is Y_{X=0}—their outcome had they received X=0.
• Answering counterfactuals requires a fully specified Structural Causal Model (SCM), not just a graph, as it involves reasoning about the specific values of latent noise variables for that individual.