Chain graphs for learning

Chain graphs combine directed and undi­ rected graphs and their underlying mathe­ matics 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, cluster­ ing with conditional interaction between vari­ ables, 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]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[2]  A. Dawid.  Conditional Independence in Statistical Theory , 1979 .

[3]  Ross D. Shachter.  Evaluating Influence Diagrams , 1986, Oper. Res..

[4]  David J. Spiegelhalter,et al.  Local computations with probabilities on graphical structures and their application to expert systems , 1990 .

[5]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .

[6]  N. Wermuth,et al.  Graphical Models for Associations between Variables, some of which are Qualitative and some Quantitative , 1989 .

[7]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.

[8]  D. Geman.  Random fields and inverse problems in imaging , 1990 .

[9]  Judea Pearl,et al.  Equivalence and Synthesis of Causal Models , 1990, UAI.

[10]  Ross D. Shachter.  An ordered examination of influence diagrams , 1990, Networks.

[11]  A. P. Dawid,et al.  Independence properties of directed Markov fields. Networks, 20, 491-505 , 1990 .

[12]  J. N. R. Jeffers,et al.  Graphical Models in Applied Multivariate Statistics. , 1990 .

[13]  N. Wermuth,et al.  On Substantive Research Hypotheses, Conditional Independence Graphs and Graphical Chain Models , 1990 .

[14]  M. Frydenberg.  The chain graph Markov property , 1990 .

[15]  Steffen L. Lauritzen,et al.  Independence properties of directed markov fields , 1990, Networks.

[16]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[17]  Michael P. Wellman,et al.  Planning and Control , 1991 .

[18]  Dan Geiger,et al.  An Entropy-based Learning Algorithm of Bayesian Conditional Trees , 1992, UAI.

[19]  Radford M. Neal.  Connectionist Learning of Belief Networks , 1992, Artif. Intell..

[20]  David J. Spiegelhalter,et al.  Bayesian analysis in expert systems , 1993 .

[21]  A. Dawid,et al.  Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models , 1993 .

[22]  R. Palmer,et al.  Introduction to the theory of neural computation , 1994, The advanced book program.

[23]  Walter R. Gilks,et al.  A Language and Program for Complex Bayesian Modelling , 1994 .

[24]  Wray L. Buntine.  Operations for Learning with Graphical Models , 1994, J. Artif. Intell. Res..

[25]  B. Thiesson,et al.  BIFROST { Block recursive models Induced From Relevant knowledge, Observations, and Statistical Techniques , 1992 .

[26]  S. Lauritzen.  The EM algorithm for graphical association models with missing data , 1995 .

[27]  E. H. Mamdani,et al.  Real-World Applications of Bayesian Networks - Introduction. , 1995 .

[28]  Michael P. Wellman,et al.  Real-world applications of Bayesian networks , 1995, CACM.

[29]  D. Madigan,et al.  On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs , 1997 .