Bayesian MAP model selection of chain event graphs.

*(English)*Zbl 1216.62039Summary: Chain event graphs are graphical models that while retaining most of the structural advantages of Bayesian networks for model interrogation, propagation and learning, more naturally encode asymmetric state spaces and the order in which events happen than Bayesian networks do. In addition, the class of models that can be represented by chain event graphs for a finite set of discrete variables is a strict superset of the class that can be described by Bayesian networks. In this paper we demonstrate how with complete sampling, conjugate closed form model selection based on product Dirichlet priors is possible, and prove that suitable homogeneity assumptions characterise the product Dirichlet prior on this class of models. We demonstrate our techniques using two educational examples.

Reviewer: Reviewer (Berlin)

##### MSC:

62F15 | Bayesian inference |

68T35 | Theory of languages and software systems (knowledge-based systems, expert systems, etc.) for artificial intelligence |

05C90 | Applications of graph theory |

62-07 | Data analysis (statistics) (MSC2010) |

PDF
BibTeX
XML
Cite

\textit{G. Freeman} and \textit{J. Q. Smith}, J. Multivariate Anal. 102, No. 7, 1152--1165 (2011; Zbl 1216.62039)

Full Text:
DOI

##### References:

[1] | Bernardo, J.M., Reference posterior distributions for Bayesian inference, Journal of the royal statistical society. series B (methodological), 41, 2, 113-147, (1979) · Zbl 0428.62004 |

[2] | Bernardo, J.; Smith, A.F.M., Bayesian theory, (1994), Wiley Chichester, England |

[3] | Boutilier, C.; Friedman, N.; Goldszmidt, M.; Koller, D., Context-specific independence in Bayesian networks, (), 115-123 |

[4] | R. Castelo, The discrete acyclic digraph Markov model in data mining, Ph.D. Thesis, Faculteit Wiskunde en Informatica, Universiteit Utrecht, April 2002. |

[5] | Cowell, R.G.; Dawid, A.P.; Lauritzen, S.L.; Spiegelhalter, D.J., Probabilistic networks and expert systems, (1999), Springer · Zbl 0937.68121 |

[6] | Cussens, J., Bayesian network learning by compiling to weighted MAX-SAT, (), 105-112 |

[7] | Denison, D.G.T.; Holmes, C.C.; Mallick, B.K.; Smith, A.F.M., () |

[8] | Geiger, D.; Heckerman, D., A characterization of the Dirichlet distribution through global and local parameter independence, The annals of statistics, 25, 3, 1344-1369, (1997) · Zbl 0885.62009 |

[9] | Heard, N.A.; Holmes, C.C.; Stephens, D.A., A quantitative study of gene regulation involved in the immune response of anopheline mosquitoes: an application of Bayesian hierarchical clustering of curves, Journal of the American statistical association, 101, 473, 18-29, (2006) · Zbl 1118.62368 |

[10] | Heckerman, D., A tutorial on learning with Bayesian networks, (), 301-354 · Zbl 0921.62029 |

[11] | Heckerman, D.; Wellman, M.P., Bayesian networks, Communications of the ACM, 38, 3, 27-30, (1995) |

[12] | Jeffreys, H., An invariant form for the prior probability in estimation problems, Proceedings of the royal society of London, series A (mathematical and physical sciences), 186, 1007, 453-461, (1946) · Zbl 0063.03050 |

[13] | Kotz, S.; Balakrishnan, N.; Johnson, N.L., () |

[14] | Lau, J.W.; Green, P.J., Bayesian model-based clustering procedures, Journal of computational and graphical statistics, 16, 3, 526-558, (2007) |

[15] | Lauritzen, S.L., () |

[16] | Poole, D.; Zhang, N.L., Exploiting contextual independence in probabilistic inference, Journal of artificial intelligence research, 18, 263-313, (2003) · Zbl 1056.68144 |

[17] | Richardson, S.; Green, P.J., On Bayesian analysis of mixtures with an unknown number of components, Journal of the royal statistical society. series B (methodological), 59, 4, 731-792, (1997) · Zbl 0891.62020 |

[18] | Shafer, G., The art of causal conjecture, artificial intelligence, (1996), The MIT Press |

[19] | Silander, T.; Kontkanen, P.; MyllymĂ¤ki, P., On sensitivity of the MAP Bayesian network structure to the equivalent sample size parameter, (), 360-367 |

[20] | Smith, J.Q.; Anderson, P.E., Conditional independence and chain event graphs, Artificial intelligence, 172, 1, 42-68, (2008) · Zbl 1182.68303 |

[21] | Smith, J.E.; Holtzman, S.; Matheson, J.E., Structuring conditional relationships in influence diagrams, Operations research, 41, 2, 280-297, (1993) |

[22] | Spiegelhalter, D.J.; Lauritzen, S.L., Sequential updating of conditional probabilities on directed graphical structures, Networks, 20, 5, 579-605, (1990) · Zbl 0697.90045 |

[23] | Steck, H.; Jaakkola, T., On the Dirichlet prior and Bayesian regularization, (), 697-704 |

[24] | Thwaites, P.; Smith, J.Q.; Riccomagno, E., Artificial intelligence, 174, 12-13, 889-909, (2010) |

[25] | Weatherburn, C.E., A first course in mathematical statistics, (1949), CUP · Zbl 0060.29503 |

[26] | West, D.B., Introduction to graph theory, (2001), Pearson Education Asia Limited and China Machine Press China |

[27] | Wilks, S.S., Mathematical statistics, (1962), Wiley New York · Zbl 0173.45805 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.