Abstract

Chain graphs combine directed and undirected graphs and their underlying mathematics combines properties of the two. This paper gives a simplified definition of chain graphs based on a hierarchical combination of Bayesian (directed) and Markov (undirected) networks. Examples of a chain graph are multivariate feed-forward networks, clustering with conditional interaction between variables, and forms of Bayes classifiers. Chain graphs are then extended using the notation of plates so that samples and data analysis problems can be represented in a graphical model as well. Implications for learning are discussed in the conclusion. 1 Introduction Probabilistic networks are a notational device that allow one to abstract forms of probabilistic reasoning without getting lost in the mathematical detail of the underlying equations. They offer a framework whereby many forms of probabilistic reasoning can be combined and performed on probabilistic models without careful hand programming. Efforts ...

Citations

5664	Probabilistic Reasoning in Intelligent Systems - Pearl - 1988
3340	Pattern Classification and Scene Analysis - Duda, Hart - 1973
1530	Introduction to the Theory of Neural Computation - Hertz, Krogh, et al. - 1991
1081	Local Computations with Probabilities on graphical structures and their applications to expert systems (with discussion - Lauritzen, Spiegelhalter
316	Gaussian influence diagrams - Shachter, Kenley - 1989
288	Planning and Control - Dean, Wellman - 1991
213	Operations for learning with graphical models - Buntine - 1994
179	The EM algorithm for graphical association models with missing data - Lauritzen - 1995
179	Equivalence and synthesis of causal models - Verma, Pearl
177	Bayesian analysis in expert systems - Spiegelhalter, Dawid, et al. - 1993
174	Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 43(1):1–59 - Besag, York, et al. - 1991
160	Connectionist learning of belief networks - Neal - 1992
147	Conditional independence in statistical theory - Dawid - 1979
145	Graphical models for the association between variables, some of which are qualitative and some - Lauritzen, Wermuth - 1989
125	Independence properties of directed Markov fields - Lauritzen, Dawid, et al. - 1990
100	Hyper markov laws in the statistical analysis of decomposable graphical models - Dawid, Lauritzen - 1993
91	A language and program for complex Bayesian modelling - Gilks, Thomas, et al. - 1993
83	The chain graph Markov property - Frydenberg - 1990
51	Random fields and inverse problems in imaging - Geman - 1990
49	Real-world applications of Bayesian networks: Introduction - Heckerman, Mamdani, et al. - 1995
41	On substantive research hypotheses, conditional independence graphs and graphical chain models - Wermuth, Lauritzen - 1989
28	An entropy-based learning algorithm of Bayesian conditional trees - Geiger
23	On the Markov equivalence of chain graphs, undirected graphs, and acyclic digraphs - Andersson, Madigan, et al. - 1997
21	An Ordered Examination of Influence Diagrams - Shachter - 1990
19	Thinking backwards for knowledge acquisition - Shachter, Heckerman - 1987
14	BIFROST --- Block recursive models Induced From Relevant knowledge - H��jsgaard - 1992
8	An in uence diagram approach to medical technology assessment - Shachter, Eddy, et al. - 1990
6	Network methods in statistics - Ripley - 1994
4	Bayesian networks for knowledge representation and learning", in Advances in Knowledge Discovery and Data - Heckerman - 1995