## Graphical ModelsThe idea of modelling systems using graph theory has its origin in several scientific areas: in statistical physics (the study of large particle systems), in genetics (studying inheritable properties of natural species), and in interactions in contingency tables. The use of graphical models in statistics has increased considerably over recent years and the theory has been greatly developed and extended. This book provides the first comprehensive and authoritative account of the theory of graphical models and is written by a leading expert in the field. It contains the fundamental graph theory required and a thorough study of Markov properties associated with various type of graphs. The statistical theory of log-linear and graphical models for contingency tables, covariance selection models, and graphical models with mixed discrete-continous variables in developed detail. Special topics, such as the application of graphical models to probabilistic expert systems, are described briefly, and appendices give details of the multivarate normal distribution and of the theory of regular exponential families. The author has recently been awarded the RSS Guy Medal in Silver 1996 for his innovative contributions to statistical theory and practice, and especially for his work on graphical models. |

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### Contents

1 | |

2 | |

4 | |

7 | |

213 Simplicial subsets and perfect sequences | 13 |

214 Subgraphs of decomposable graphs | 19 |

22 Hypergraphs | 21 |

222 Graphs and hypergraphs | 22 |

62 Graphical interaction models | 173 |

622 Maximum likelihood estimation | 175 |

63 Decomposable models | 187 |

632 Maximum likelihood estimation | 188 |

633 Exact tests in decomposable models | 191 |

64 Hierarchical interaction models | 199 |

642 Generators and canonical statistics | 201 |

643 Maximum likelihood estimation | 205 |

223 Junction trees and forests | 24 |

23 Notes | 26 |

Conditional independence and Markov properties | 28 |

32 Markov properties | 32 |

322 Markov properties on directed acyclic graphs | 46 |

323 Markov properties on chain graphs | 53 |

33 Notes | 60 |

Contingency tables | 62 |

422 Saturated models | 70 |

423 Logaffine and loglinear models | 71 |

43 Hierarchical models | 81 |

431 Estimation in hierarchical logafrine models | 82 |

432 Test in hierarchical models | 85 |

433 Interaction graphs and graphical models | 88 |

44 Decomposable models | 90 |

442 Maximum likelihood estimation | 91 |

443 Exact tests in decomposable models | 98 |

444 Asymptotic tests in decomposable models | 105 |

45 Recursive models | 106 |

451 Recursive graphical models | 107 |

452 Recursive hierarchical models | 112 |

46 Blockrecursive models | 113 |

461 Chain graph models | 114 |

462 Blockrecursive hierarchical models | 118 |

463 Decomposable blockrecursive models | 119 |

Multivariate normal models | 123 |

512 The saturated model | 124 |

513 Conditional independence | 129 |

514 Interaction | 131 |

521 Maximum likelihood estimation | 132 |

522 Deviance tests | 142 |

53 Decomposable models | 144 |

532 Maximum likelihood estimation | 145 |

533 Exact tests in decomposable models | 149 |

54 Notes | 153 |

542 Lattice models | 156 |

Models for mixed data | 158 |

612 The saturated models | 168 |

644 Mixed hierarchical model subspaces | 213 |

65 Chain graph models | 216 |

651 CG regressions | 217 |

652 Estimation in chain graph models | 218 |

66 Notes | 219 |

662 Bibliographical notes | 220 |

Further topics | 221 |

711 Specification of the joint distribution | 223 |

712 Local computation algorithm | 226 |

713 Extensions | 228 |

72 Model selection | 229 |

73 Modelling complexity | 230 |

731 Markov chain Monte Carlo methods | 231 |

732 Applications | 232 |

74 Missingdata problems | 233 |

742 Hierarchical loglinear models | 234 |

743 Recursive models | 235 |

Various prerequisites | 237 |

A2 KullbackLeibler divergence | 238 |

A3 Mobius inversion | 239 |

A 5 Sufficiency | 241 |

Linear algebra and random vectors | 243 |

B2 Factor subspaces and interactions | 246 |

B3 Random vectors | 250 |

The multivariate normal distribution | 254 |

C2 The Wishart distribution | 258 |

C3 Other derived distributions | 262 |

C32 Wilkss distribution | 263 |

C33 Test for identical covariances | 264 |

Exponential models | 266 |

D12 Analytic properties | 267 |

D13 Maximum likelihood estimation | 268 |

D15 Iterative computational methods | 269 |

D2 Curved exponential models | 272 |

D22 The singular case | 276 |

278 | |

295 | |

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### Common terms and phrases

algorithm assume asymptotic block-recursive calculated canonical parameter canonical statistic CG density CG distribution chain components chain graph model cliques of Q complete concentration matrix conditional distribution conditional independence consider continuous variables corresponding counts covariance matrix covariance selection model decomposable graph decomposable models defined denote deviance test direct join directed acyclic graph discrete variables edge equivalent example factorization follows given graph Q graphical models Hence hierarchical model holds homogeneous hyperedges hypergraph implies inverse iterative Lauritzen Lemma likelihood equations likelihood function likelihood ratio linear log-affine model log-linear models marginal table marked graph Markov property maximum likelihood estimate model with graph moral graph multinomial sampling multivariate normal nd(a normal distribution notation obtained pairwise partitioned perfect sequence positive definite probability Proof Proposition quadratic random variables regular exponential model restrictions result satisfies saturated model Section simplicial space subgraph subsets sufficient statistic Theorem triangulated undirected graph vertex vertices zero