Also known as Belief Network, Causal Network.
A Bayesian Network (BN) 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 BN 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). Existence of an arc between two nodes indicates that the parent node has a direct impact on the descendant node. A missing arc between nodes means those nodes are conditionally dependent only. The BN represents two types of knowledge:
- Qualitative knowledge: structure of the graph (nodes with arcs)
- Quantitative knowledge: CPT – conditional probability tables (discrete domain) or CPD – conditional probability distributions (continuous domain).
Formal definition of the BN:
which is a factoring formula (chaining rule) used to compute the joint probability distribution.
An example: Bayesian belief network
Other concepts contained in the BN models:
- Various types of evidence: soft, hard
- Various types of nodes connection: linear (serial, cascade), divergent, convergent
- Active path and d-separation: depending on a type of the causal chain (causal chains, common causes, common consequences)
- Various types of inference: diagnostic, causal, intercausal (so called explaining away), and mixed
In real applications we can meet various specialized nodes such as Noisy-AND, Noisy-OR, Leaky-Noisy-AND, Noisy-MAX, etc. that help us to reduce a number of probabilities in CPTs to be defined, to use multivalued discrete nodes, to model sequential dependency of nodes, etc. Adding additional types of nodes (utility/functional nodes and decision nodes) into the graph of the BN with event nodes, we can transform the BN into the decision network (also called Influence Diagram). Another specialized type of the BN is a Dynamic BN (also Two-Timeslice BN, 2TBN) which relates variables to each other over adjacent time steps; Object-oriented BN (OOBN) that has nodes that represent instances of other network fragments producing a hierarchical model, etc.
Literature
- Pearl Judea, Probabilistic reasoning in intelligent systems: Networks of plausible inference. Morgan Kaufmann, 1988
- Pearl Judea, Causality – Models, Reasoning, and Inference (2nd ed.). Cambridge University Press, 2009