Bayesian network

A Bayesian network consists of a directed acyclic graph (DAG) together with a probability model for its variables. Each node represents a random variable, and the directed edges define which variables are parents of which others. A node’s conditional probability distribution (or, for discrete variables, conditional probability table) specifies its probability given its parents; a node without parents has a prior distribution. The graph encodes conditional-independence assumptions about the joint distribution.

For variables X₁, …, Xₙ in a Bayesian network, the joint distribution factors into the product, over all variables, of each variable’s conditional probability given its parent variables. This representation can require far fewer probabilities than listing every possible joint outcome, particularly when each variable has relatively few parents. The graph’s structure also identifies conditional independencies: certain variables become independent once specified other variables are known. The validity and usefulness of these simplifications depend on the model’s structure and assumptions.

Inference uses the network to calculate probabilities of interest, including updated probabilities after evidence about one or more variables is observed. For example, a model might estimate the probability of rain after observing wet grass, while accounting for other possible explanations such as a sprinkler. Networks can be constructed from expert knowledge, learned from data, or developed using both. Learning their structure and conditional distributions involves choices about data, assumptions, and methods; a learned graph is not automatically a validated explanation of the world.

An arrow in a Bayesian network indicates a directed dependency in the model, but does not by itself prove that one variable causes another. A causal interpretation requires additional knowledge and assumptions, such as a defensible causal structure and consideration of confounding factors that may not have been measured. Observational data can be consistent with multiple graph structures that imply the same probability distribution. For questions about interventions—what would happen if a variable were deliberately changed—causal assumptions must be made explicit and justified.

Bayesian networks are used to represent uncertain relationships and support reasoning in fields such as medicine, engineering, environmental science, and artificial intelligence. They can combine data with domain knowledge and make assumptions visible in a graphical form. Their practical limits include the difficulty of selecting a suitable structure, estimating reliable probabilities when data are sparse, and carrying out inference in large or densely connected networks. Results are conditional on the chosen model and should be interpreted in light of its assumptions and evidence.