A probabilistic network, which is sometimes referred to as a probabilistic graphical model, is a tool that enables us to visually illustrate and work with conditional independencies among variables in a given problem. Specifically, nodes represent variables and the lack of an edge between two nodes represents conditional independence between the variables. There are two types of graphical models: directed graphical models and undirected graphical models. Undirected graphical models are called Markov random fields or Markov networks and are popularly used in the physics and vision communities. Directed graphical models have no directed cycles and are called Bayesian networks (BNs) or belief networks; they are popularly used in the artificial intelligence and statistics communities.
In an undirected graphical model, two nodes A and B are defined to be conditionally independent given a third node C, written A ⊥ B|C, if all paths between A and B are separated by C. If the joint distribution of A, B, and C is known, then we may write:
In a directed graphical model, conditional independence can be displayed graphically. For example, consider the distribution:
For each conditional distribution we add a directed arc from the node corresponding to the ...