A graph in which nodes represent random variables. Pairs of nodes connected by arcs correspond to variables that are not independent of one another. If two sets of nodes A and B are not connected, except via paths that pass through a node in a third set, C, then the variables represented by the nodes in the set A are conditionally independent of the variables represented by the nodes in the set B, given the values of the variables represented by the nodes in the set C.

If an arc has no direction attached, then this implies that there is association but not causation. An undirected graphical model (also called a Markov random field) is a model in which no arcs have directions attached. A model in which all arcs have directions attached is called a directed graphical model (or Bayes network or Bayes net). If we consider a directed path as referring to ‘generations’ of variables, then a simple example of conditional independence (not the only possibility) occurs when a ‘child variable’ is conditionally independent of earlier ‘ancestor variables’, given the values of the ‘parent variables’. A special case is the hidden Markov model. A directed graphical model having no path that leads from a node back to that same node is termed a directed acyclic graph. Such graphs occur in expert systems. A graph containing a mixture of directed and undirected arcs is a chain graph.