## Learning in Graphical ModelsIn the past decade, a number of different research communities within the computational sciences have studied learning in networks, starting from a number of different points of view. There has been substantial progress in these different communities and surprising convergence has developed between the formalisms. The awareness of this convergence and the growing interest of researchers in understanding the essential unity of the subject underlies the current volume. Two research communities which have used graphical or network formalisms to particular advantage are the belief network community and the neural network community. Belief networks arose within computer science and statistics and were developed with an emphasis on prior knowledge and exact probabilistic calculations. Neural networks arose within electrical engineering, physics and neuroscience and have emphasised pattern recognition and systems modelling problems. This volume draws together researchers from these two communities and presents both kinds of networks as instances of a general unified graphical formalism. The book focuses on probabilistic methods for learning and inference in graphical models, algorithm analysis and design, theory and applications. Exact methods, sampling methods and variational methods are discussed in detail. Audience: A wide cross-section of computationally oriented researchers, including computer scientists, statisticians, electrical engineers, physicists and neuroscientists. |

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### Contents

ADVANCED INFERENCE IN BAYESIAN NETWORKS | 27 |

INFERENCE IN BAYESIAN NETWORKS USING NESTED JUNCTION TREES | 51 |

A UNIFYING FRAMEWORK FOR PROBABILISTIC INFERENCE | 75 |

AN INTRODUCTION TO VARIATIONAL METHODS FOR GRAPHICAL MODELS | 105 |

IMPROVING THE MEAN FIELD APPROXIMATION VIA THE USE OF MIXTURE DISTRIBUTIONS | 163 |

INTRODUCTION TO MONTE CARLO METHODS | 175 |

SUPPRESSING RANDOM WALKS IN MARKOV CHAIN MONTE CARLO USING ORDERED OVERRELAXATION | 205 |

CHAIN GRAPHS AND SYMMETRIC ASSOCIATIONS | 231 |

DATA CLUSTERING AND DATA VISUALIZATION | 405 |

LEARNING BAYESIAN NETWORKS WITH LOCAL STRUCTURE | 421 |

ASYMPTOTIC MODEL SELECTION FOR DIRECTED NETWORKS WITH HIDDEN VARIABLES | 461 |

A HIERARCHICAL COMMUNITY OF EXPERTS | 479 |

AN INFORMATIONTHEORETIC ANALYSIS OF HARD AND SOFT ASSIGNMENT METHODS FOR CLUSTERING | 495 |

LEARNING HYBRID BAYESIAN NETWORKS FROM DATA | 521 |

A MEAN FIELD LEARNING ALGORITHM FOR UNSUPERVISED NEURAL NETWORKS | 541 |

EDGE EXCLUSION TESTS FOR GRAPHICAL GAUSSIAN MODELS | 555 |

THE MULTIINFORMATION FUNCTION AS A TOOL FOR MEASURING STOCHASTIC DEPENDENCE | 261 |

A TUTORIAL ON LEARNING WITH BAYESIAN NETWORKS | 301 |

A VIEW OF THE EM ALGORITHM THAT JUSTIFIES INCREMENTAL SPARSE AND OTHER VARIANTS | 355 |

LATENT VARIABLE MODELS | 371 |

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### Common terms and phrases

acyclic algorithm approach approximation Artificial Intelligence assignment Bayesian networks belief networks Boltzmann machine bucket causal chain graph clique clustering components compute conditional distribution conditional independence conditional probability configuration consider convergence corresponding covariance CPDs data point data set defined denote density dependence directed graphs discrete edges EM algorithm encoding entropy Equation estimate evaluate example factor function Gaussian process Gibbs sampling given graphical models Heckerman hidden variables independency model inference rule iterations joint probability Jordan junction tree KL divergence latent variable Lauritzen learning Lemma linear units lower bound marginal marginal likelihood Markov chain Markov property matrix maximizes mean field Monte Carlo methods moral graph Morgan Kaufmann multiinformation network structure neural networks nodes observed obtain optimal ordered overrelaxation parents partition prior probability distribution problem procedure random variables representation score space Spiegelhalter statistics subset tion Uncertainty in Artificial undirected update values variance vector