Causal Bayesian Network

The foundational structure of a CBN is a Directed Acyclic Graph (DAG) where nodes represent random variables and directed edges (→) represent assumed direct causal relationships. The 'acyclic' property ensures no variable can be a cause of itself. This graph encodes qualitative causal assumptions, such as Smoking → Tar in Lungs → Cancer. Unlike a standard Bayesian network, the directionality in a CBN is interpreted as a physical or mechanistic causal influence, not merely a statistical dependency.

Each node in the CBN is governed by a structural equation. For a variable X with parents Pa(X), the equation is: X := f(Pa(X), U_X). Here, := denotes assignment, f is a deterministic function, and U_X is an independent noise or exogenous variable representing unmodeled causes. This set of equations forms the Structural Causal Model (SCM), the mathematical engine behind the graph. It defines how each variable is generated from its direct causes, enabling the simulation of interventions by modifying these equations.

This condition links the causal graph to probability. It states a variable is conditionally independent of its non-descendants given its direct causes (parents). This allows the joint probability distribution P over all variables (V1, V2, ..., Vn) to factorize according to the graph structure: P(V1, V2, ..., Vn) = Π P(Vi | Pa(Vi)). This factorization is identical to a standard Bayesian network, but in a CBN, the parents are causal parents. This is the bridge that allows probabilistic inference from observational data under the causal assumptions.

The key operator that distinguishes causal from statistical reasoning. The do-operator, do(X=x), represents an external intervention that sets variable X to value x, irrespective of its natural causes. In the SCM, this means replacing the equation for X with X := x. The resulting distribution, P(Y | do(X=x)), is the interventional distribution. It answers questions like "What would the probability of cancer be if we forced everyone to smoke?" This is computed by the truncated factorization or g-formula, which removes the term P(X | Pa(X)) from the joint factorization.

The most advanced level of reasoning supported by a fully-specified CBN (with functional SCMs). A counterfactual queries what would have happened in the same unit (e.g., a specific patient) under a different hypothetical past. It answers "What would John's cancer outcome have been, had he not smoked, given that he did smoke and did get cancer?" Computation requires three steps: 1) Abduction: Infer the specific noise values U for the unit given the observed evidence. 2) Action: Modify the model with the intervention (do(no smoke)). 3) Prediction: Compute the outcome using the updated model and the inferred U.

Not all causal queries can be answered from observational data alone. Causal identifiability is the property that a causal effect P(Y | do(X)) can be uniquely computed from the observed distribution P and the causal graph. do-Calculus, developed by Judea Pearl, provides a complete set of rules to transform interventional probabilities into observational probabilities when identifiability holds. It uses graphical criteria like the backdoor criterion (to block confounding paths) and the frontdoor criterion (to use mediators) to determine if and how an effect can be estimated from passive data.

A Structural Causal Model (SCM) is the formal mathematical foundation for a Causal Bayesian Network. 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.
• An associated causal graph (a DAG) that visually represents these dependencies.
• While a CBN provides the probabilistic semantics, the SCM provides the algebraic 'machinery' necessary for computing interventions and counterfactuals, the higher rungs of the causal hierarchy.

Do-calculus is a set of three inference rules that allow researchers to answer causal questions from a combination of observational data and a known causal graph. Its primary function is to transform expressions containing the do-operator—which represents an intervention—into expressions involving only standard observational probabilities. This is the mathematical engine that enables a Causal Bayesian Network to compute the effects of hypothetical actions or policies without physically running an experiment.

Causal identifiability is the fundamental property that determines whether a causal query (e.g., the Average Treatment Effect) can be uniquely computed from available data under a given causal model. Before any estimation occurs, one must check if the effect is identifiable. For Causal Bayesian Networks, graphical criteria like the backdoor criterion and frontdoor criterion provide tests for identifiability. If an effect is not identifiable, no statistical method can estimate it without making additional, untestable assumptions.

Causal discovery refers to algorithms that attempt to learn the causal graph itself from data, providing the structure needed for a Causal Bayesian Network. Key methods include:

• Constraint-based algorithms (e.g., PC, FCI): Test for conditional independencies to infer edge presence and orientation.
• Score-based algorithms: Search the space of DAGs to find the structure that best fits the data according to a scoring function.
• These algorithms rely on assumptions like the Causal Markov Condition and Causal Faithfulness to make valid inferences from observational data.

The backdoor criterion is a graphical rule used to find a sufficient set of variables to adjust for in order to estimate a causal effect from observational data. A set of variables Z satisfies the backdoor criterion for estimating the effect of X on Y if:

• Z blocks every backdoor path (a non-causal path connecting X and Y that starts with an arrow into X).
• No node in Z is a descendant of X. Conditioning on such a Z 'closes' all spurious, non-causal paths, allowing the causal effect to be identified via standard statistical adjustment (e.g., regression, matching).

The Causal Markov Condition is the core assumption linking causal structure to probability. It states that in a Causal Bayesian Network, a variable is conditionally independent of all its non-descendants in the graph, given its direct causes (its parents). This assumption justifies reading conditional independence relationships directly from the causal graph using d-separation. It is essential for causal discovery algorithms and for simplifying the joint probability distribution represented by the network into the product of local conditional probabilities.