## Graphical Models: Foundations of Neural ComputationGraphical models use graphs to represent and manipulate joint probability distributions. They have their roots in artificial intelligence, statistics, and neural networks. The clean mathematical formalism of the graphical models framework makes it possible to understand a wide variety of network-based approaches to computation, and in particular to understand many neural network algorithms and architectures as instances of a broader probabilistic methodology. It also makes it possible to identify novel features of neural network algorithms and architectures and to extend them to more general graphical models.This book exemplifies the interplay between the general formal framework of graphical models and the exploration of new algorithms and architectures. The selections range from foundational papers of historical importance to results at the cutting edge of research.Contributors H. Attias, C. M. Bishop, B. J. Frey, Z. Ghahramani, D. Heckerman, G. E. Hinton, R. Hofmann, R. A. Jacobs, Michael I. Jordan, H. J. Kappen, A. Krogh, R. Neal, S. K. Riis, F. B. Rodr guez, L. K. Saul, Terrence J. Sejnowski, P. Smyth, M. E. Tipping, V. Tresp, Y. Weiss. |

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### Contents

Probabilistic Independence Networks for Hidden Markov Probability Models | 1 |

Learning and Relearning in Boltzmann Machines | 45 |

Learning in Boltzmann Trees | 77 |

Deterministic Boltzmann Learning Performs Steepest Descent in WeightSpace | 89 |

Attractor Dynamics in Feedforward Neural Networks | 97 |

Efficient Learning in Boltzmann Machines Using Linear Response Theory | 121 |

Asymmetric Parallel Boltzmann Machines are Belief Networks | 141 |

Variational Learning in Nonlinear Gaussian Belief Networks | 145 |

Independent Factor Analysis | 207 |

Hierarchical Mixtures of Experts and the EM Algorithm | 257 |

Hidden Neural Networks | 291 |

Variational Learning for Switching StateSpace Models | 315 |

Nonlinear TimeSeries Prediction with Missing and Noisy Data | 349 |

Correctness of Local Probability Propagation in Graphical Models with Loops | 367 |

409 | |

Mixtures of Probabilistic Principal Component Analyzers | 167 |