In statistics and

Markov model In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Mark ...

ing, an ancestral graph is a type of

mixed graph In graph theory, a mixed graph is a graph consisting of a set of vertices , a set of (undirected) edges , and a set of directed edges (or arcs) . Definitions and notation Consider adjacent vertices u,v \in V. A directed edge, called an arc, ...

to provide a graphical representation for the result of marginalizing one or more vertices in a

graphical model A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability ...

that takes the form of a

directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one v ...

Definition

Ancestral graphs are mixed graphs used with three kinds of edges: directed edges, drawn as an arrow from one vertex to another, bidirected edges, which have an arrowhead at both ends, and undirected edges, which have no arrowheads. It is required to satisfy some additional constraints: *If there is an edge from a vertex ''u'' to another vertex ''v'', with an arrowhead at ''v'' (that is, either an edge directed from ''u'' to ''v'' or a bidirected edge), then there does not exist a path from ''v'' to ''u'' consisting of undirected edges and/or directed edges oriented consistently with the path. *If a vertex ''v'' is an endpoint of an undirected edge, then it is not also the endpoint of an edge with an arrowhead at ''v''.

Applications

Ancestral graphs are used to depict conditional independence relations between variables in Markov models.

References

{{reflist Extensions and generalizations of graphs Graphical models