A probabilistic graphical model represented by a directed acyclic graph where nodes are random variables and edges express conditional dependence used to factorize a joint probability.

Principle

Encodes conditional independences so the joint probability decomposes into a product of local conditional probabilities, enabling modular specification and inference.

Demonstration

A diagnostic model with nodes for disease, test result, and symptom uses directed edges from disease to symptom and test; the joint factorizes into P(disease) P(test|disease) P(symptom|disease).

Misapplication

Treating a system with feedback loops as a Bayesian network without converting or approximating the cycles violates the acyclicity requirement and misrepresents dependencies.

Consequence

Supports efficient local computation for likelihoods and marginalization, and clarifies which observations render other variables independent given measured parents.

Reversal

An undirected graphical model: represents relations by symmetric adjacency and encodes independence through graph separation rather than directed conditional factorization.

Boundary

Requires a directed acyclic structure; continuous or discrete variables both allowed but parameterization and inference methods differ across domains and likelihood forms.

Semantic Tension

Often contrasted with causal models: a Bayesian network can represent causal assumptions but, by itself, is a probabilistic factorization that does not prove causation without further constraints.

Synthesis

A Bayesian network is a DAG-structured probabilistic model that factorizes a joint distribution into local conditional factors, making dependencies explicit and inference tractable.