Structural Causal Model (SCM)

The set of endogenous variables (V) represent the observed or latent quantities of interest in the system. Each variable is defined by a structural equation that specifies its value as a deterministic function of its direct causes (parents) and an independent exogenous noise term (U). This formalizes the notion that each variable is generated by its causal parents plus random, unexplained variation.

These are the background variables or noise terms that represent all unmodeled, external factors influencing the endogenous variables. Each U is:

• Assigned to one or more endogenous variables.
• Assumed to be mutually independent.
• The source of randomness and uncertainty in the model. The joint distribution P(U) over these variables, combined with the structural equations, fully determines the model's behavior and the resulting observational distribution P(V).

The core mathematical component. For each variable V_i, there is a function f_i that defines it:

V_i := f_i(PA_i, U_i)

Where PA_i are the direct causes (parents) of V_i in the causal graph. These equations are non-parametric and asymmetric, representing assignment, not mere association. They encode the data-generating process. For example, in a simple model: Sales := f(Advertising_Budget, Economic_Climate, U_Sales).

A directed acyclic graph (DAG) that provides a visual and mathematical representation of the causal assumptions. Each node is a variable in V. A directed edge from X to Y means X is a direct cause of Y (i.e., X appears in the structural equation for Y). The graph encodes conditional independence relationships via d-separation, which, under the Causal Markov Condition, are reflected in the observed data. This graph is the blueprint for reasoning about interventions and counterfactuals.

The do-operator, denoted do(X=x), is a key semantic element of an SCM. It represents an external intervention that sets variable X to value x, overriding its natural structural equation. In the graph, this is modeled by deleting all incoming edges to X. The SCM allows computation of the interventional distribution P(V | do(X=x)), answering "what if" questions. This formally distinguishes seeing (P(Y|X=x)) from doing (P(Y|do(X=x))).

The highest level of reasoning enabled by a fully-specified SCM (including the functional forms of F and distribution of U). A counterfactual asks a question about a specific unit under hypothetical, contrary-to-fact conditions (e.g., "Would this patient have survived if they had not received the drug?"). Answering requires:

1. Abduction: Infer the likely noise values U for the unit given observed facts.
2. Action: Apply the do-operator to modify the model.
3. Prediction: Simulate the new outcome using the same inferred U. This process is uniquely enabled by the SCM's granular specification.

A causal graph is a directed acyclic graph (DAG) that visually represents the causal assumptions of an SCM. Nodes are variables, and directed edges indicate direct causal relationships. It provides the essential map for applying d-separation to read off conditional independencies and for identifying valid adjustment sets using criteria like the backdoor criterion.

Do-calculus is a formal system of three inference rules developed by Judea Pearl. It allows researchers to compute the effects of interventions (expressed with the do-operator) from purely observational data and a known causal graph. The rules mathematically transform expressions like P(Y | do(X)) into observable probabilities, enabling the answer to causal 'what if' questions without running a physical experiment.

A counterfactual query represents the highest rung on the 'ladder of causation,' answering 'what would have happened' under different circumstances. In an SCM, counterfactuals are computed by:

• Abducting the noise terms for a specific unit from observed data.
• Modifying the structural equations to reflect the hypothetical intervention.
• Propagating the new values through the model. This allows for unit-level causal reasoning, such as estimating whether a specific patient would have survived had they not received a treatment.

Causal identifiability is the fundamental property that determines whether a causal effect can be uniquely estimated from the available data and model assumptions. Before any estimation occurs, one must check if the desired quantity (e.g., Average Treatment Effect) is identifiable. This is typically verified using graphical criteria like the backdoor or frontdoor criterion, which confirm that all confounding paths can be blocked, allowing the causal effect to be expressed as a function of observable probabilities.

Causal discovery refers to a suite of algorithms that attempt to learn the causal graph (the structure of the SCM) directly from data. Methods include:

• Constraint-based algorithms (e.g., PC, FCI): Use statistical tests of conditional independence to infer edge existence and orientation.
• Score-based methods: Search the space of DAGs to find the structure that best fits the data according to a scoring function (e.g., BIC).
• Functional causal models: Leverage assumptions like non-linear relationships with additive noise to distinguish cause from effect in two-variable settings.

Invariant Causal Prediction (ICP) and related paradigms like Invariant Risk Minimization (IRM) are learning frameworks grounded in causal principles. They posit that true causal mechanisms remain invariant across different environments or contexts. The goal is to find predictors that perform consistently well across all observed environments, as these are likely based on causal parents rather on spurious, environment-specific correlations. This leads to models with superior out-of-distribution generalization.