An ancestral graph is a graph

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

 with three types of edges: directed edge, bidirected edge, and undirected edge such that it can be decomposed into three parts: an undirected subgraph, a directed subgraph, and directed edges pointing from the undirected subgraph to the directed subgraph.

An ancestral graph G over nodes V is maximal if for every pair (x, y) of non-adjacent nodes in G, there exists a set Z of nodes in V \{x, y} such that x and y are m-separated

M-separation

In statistics, m-separation is a measure of disconnectedness in ancestral graphs and a generalization of d-separation for directed acyclic graphs. It is the opposite of m-connectedness....

 by Z in G.

In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, an ancestral graph can be used to represent the statistical dependencies, and notional causative effects, between random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s.