A probabilistic graphical model in which random variables are nodes of a directed acyclic graph (DAG) whose edges represent direct causal influences; the joint distribution factorizes according to the graph via conditional probability distributions, and the graph plus probabilistic assumptions support causal inference including prediction under interventions and certain counterfactual queries.

Principle

The DAG encodes conditional independencies and causal mechanisms so that (i) the joint distribution factorizes into local conditionals and (ii) interventions are represented by modifying the graph (or conditional) to simulate external manipulation, enabling identification of causal effects under stated assumptions.

Demonstration

Situation: Variables Smoking, TarDeposition, and LungCancer form a DAG Smoking → TarDeposition → LungCancer. Recognition: The edges represent putative direct causal links and local conditional distributions are specified. Action: To evaluate the effect of reducing smoking, replace the Smoking mechanism with an intervention distribution (do(Smoking=low)) and compute the downstream distribution of LungCancer via the factorization. Consequence: The network distinguishes observational conditional probabilities (P(LungCancer | Smoking)) from interventional predictions (P(LungCancer | do(Smoking))).

Misapplication

Treating any directed graphical model learned from observational correlations as a causal DAG without justifying causal assumptions (e.g., absence of confounders, correct edge orientation), or interpreting conditional independence tests alone as proof of causal direction.

Consequence

Provides a compact formal language for encoding causal assumptions, deriving identification conditions, designing interventions and predicting the effects of manipulations; but causal conclusions require the network's structural and independence assumptions to hold or be testable.

Reversal

If the system contains feedback cycles, contemporaneous mutual causation, or pervasive unmeasured confounding, a DAG representation fails or requires extension (dynamic models, cyclic graphs, explicit latent variables); identifiability may be lost without further assumptions or experimental data.

Boundary

Within scope: causal modeling of variables amenable to intervention, identification via do‑style reasoning and graphical criteria (d‑separation, back‑door/front‑door). Outside scope: purely associational undirected graphical models that do not claim causal interpretation, and settings where no plausible intervention mapping exists.

Semantic Tension

Statistical association ↔ causal interpretation: the same probabilistic patterns can admit multiple causal explanations unless supplemented by structural assumptions or interventional data.

Synthesis

A Causal Bayesian Network is an operational encoding of causal assumptions: it compresses background knowledge into a graph whose factorization supports both observational inference and well‑defined interventional predictions, but its use requires careful articulation and testing of the causal assumptions.