In graph theory, a moral graph is used to find the equivalent undirected form of a directed acyclic graph. It is a key step of the junction tree algorithm, used in belief propagation on graphical models.
The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of nodes that have a common child, and then making all edges in the graph undirected. Equivalently, a moral graph of a directed acyclic graph G is an undirected graph in which each node of the original G is now connected to its Markov blanket. The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be married by sharing an edge.
Note that a moral graph is not necessarily chordal.
Moralization may also be applied to mixed graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two vertices that both have outgoing edges to the same chain, and then forgets the orientation of the directed edges of the graph.
- Cowell, Robert G.; Dawid, A. Philip; Lauritzen, Steffen L.; Spiegelhalter, David J. (1999). "3.2.1 Moralization". Probabilistic Networks and Expert Systems: Exact Computational Methods for Bayesian Networks. Springer-Verlag New York. pp. 31–33. doi:10.1007/0-387-22630-3_3. ISBN 0-387-98767-3.
- Cowell et al. (1999), p. 50.