M. Studeny:
Characterization of essential graphs by means of an operation of legal component merging.
In Proceedings of the 1st European Workshop on Probabilistic Graphical Models (PGM'02)
(J. A. Gamez, A. Salmeron eds.), University Castilla la Mancha 2002, pages 161-168.
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Abstract
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One of the most common ways of representing classes of equivalent Bayesian networks is the use
of essential graphs. These chain graphs are also known in the literature as completed
patterns or completed pdags. The name essential graph was proposed by Andersson,
Madigan and Perlman (1997) who also gave a graphical characterization of
essential graphs. In this contribution an alternative characterization of
essential graphs is presented. The main observation is that every essential
graph is the largest chain graph within a special class of chain graphs. More
precisely, every equivalence class of Bayesian networks is contained in an
equivalence class of chain graphs without flags (= certain induced subgraphs). A
special operation of legal merging of (connectivity) components for a
chain graph without flags is introduced. This operation leads to an algorithm
for finding the essential graph on the basis of any graph in that equivalence
class of chain graphs without flags which contains the equivalence class of a
Bayesian network. In particular, the algorithm may start with any Bayesian
network.
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AMS classification 68T30, 62H05
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Keywords
- Bayesian network
- chain graph
- essential graphs
- flag
- legal merging
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A pdf copy of
a preprint (330kB) is available.
The paper partially builds on the work:
- S. A. Andersson, D. Madigan, and M. D. Perlman:
A characterization of Markov equivalence classes for acyclic digraphs.
Annals of Statistics 25 (1997), pp. 505-541.