Steffen L. Lauritzen Elements of Graphical Models Lectures from the XXXVIth International Probability Summer School in Saint-Flour, France, 2006 September 4, 2011 Springer Your dedication goes here Preface Here come the golden words place(s), First name Surname month year First name Surname vii Contents 1 Introduction ................................................... 1 2 Markov Properties ............................................. 5 2.1 Conditional Independence . .5 2.1.1 Basic properties . .5 2.1.2 General conditional independence . .6 2.2 Markov Properties for Undirected Graphs . .9 2.3 Markov Properties for Directed Acyclic Graphs . 15 2.4 Summary . 22 3 Graph Decompositions and Algorithms ........................... 25 3.1 Graph Decompositions and Markov Properties . 25 3.2 Chordal Graphs and Junction Trees . 27 3.3 Probability Propagation and Junction Tree Algorithms. 32 3.4 Local Computation . 32 3.5 Summary . 39 4 Specific Graphical Models ...................................... 41 4.1 Log-linear Models . 41 4.1.1 Interactions and factorization . 41 4.1.2 Dependence graphs and factor graphs . 42 4.1.3 Data and likelihood function . 44 4.2 Gaussian Graphical Models . 50 4.2.1 The multivariate Gaussian distribution . 50 4.2.2 The Wishart distribution . 52 4.2.3 Gaussian graphical models . 53 4.3 Summary . 58 5 Further Statistical Theory....................................... 61 5.1 Hyper Markov Laws . 61 5.2 Meta Markov Models . 66 ix x Contents 5.2.1 Bayesian inference . 71 5.2.2 Hyper inverse Wishart and Dirichlet laws . 72 5.3 Summary . 73 6 Estimation of Structure ......................................... 77 6.1 Estimation of Structure and Bayes Factors . 77 6.2 Estimating Trees and Forests . 81 6.3 Learning Bayesian networks . 85 6.3.1 Model search methods . 85 6.3.2 Constraint-based search . 87 6.4 Summary . 88 References ......................................................... 93 Chapter 1 Introduction Conditional independence The notion of conditional independence is fundamental for graphical models. For three random variables X, Y and Z we denote this as X ??Y jZ and graphi- cally as XZYu u u If the random variables have density w.r.t. a product measure m, the conditional independence is reflected in the relation f (x;y;z) f (z) = f (x;z) f (y;z); where f is a generic symbol for the densities involved. Graphical models 2 4 @ @ 1 u 5 u 7 @@ @@ @ @ u u u @@ @@ 3u 6 u For several variables, complex systems of conditional independence can be de- scribed by undirected graphs. Then a set of variables A is conditionally independent of set B, given the values of a set of variables C if C separates A from B. 1 2 1 Introduction Has tuberculosis Has Positive X-ray? Positive Visit to Asia?to Visit Tuberculosis or canceror Tuberculosis Has lung cancer lung Has Dyspnoea? Smoker? Has bronchitis Has Fig. 1.1 An example of a directed graphical model describing the relationship between risk fac- tors, lung diseases and symptoms. This model was used by Lauritzen and Spiegelhalter (1988) to illustrate important concepts in probabilistic expert systems. A pedigree Graphical model for a pedigree from study of Werner’s syndrome. 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