A Bayesian network, Bayes network, belief network, Bayes(ian) model or probabilistic directed acyclic graphical model is a probabilistic graphical model (a type of statistical model) that represents a set of random variables and their conditional dependencies via a directed acyclic graph (DAG).

A belief network, also called a Bayesian network, is an acyclic directed graph (DAG), where the nodes are random variables. There is an arc from each element of parents (Xi) into Xi. Associated with the belief network is a set of conditional probability distributions - the conditional probability of each variable given its parents (which includes the prior probabilities of those variables with no parents).

Thus, a belief network consists of

• a DAG, where each node is labeled by a random variable;

• a domain for each random variable; and

• a set of conditional probability distributions giving P(X|parents(X)) for each variable X.

A belief network is acyclic by construction. The way the chain rule decomposes the conjunction gives the ordering. A variable can have only predecessors as parents. Different decompositions can result in different belief networks.

As part of the domain, assume the following conditional independencies:

• Fire is conditionally independent of Tampering (given no other information).

• Alarm depends on both Fire and Tampering. That is, we are making no independence assumptions about how Alarm depends on its predecessors given this variable ordering.

• Smoke depends only on Fire and is conditionally independent of Tampering and Alarm given whether there is a Fire.

• Leaving only depends on Alarm and not directly on Fire or Tampering or Smoke. That is, Leaving is conditionally independent of the other variables given Alarm.

• Report only directly depends on Leaving.

The belief network figure expresses these dependencies: