| Introduction | p. 1 |
| Motivational thoughts | p. 2 |
| Goals of the monograph | p. 6 |
| Structure of the book | p. 7 |
| Basic Concepts | p. 9 |
| Conditional independence | p. 9 |
| Semi-graphoid properties | p. 11 |
| Formal independence models | p. 11 |
| Semi-graphoids | p. 13 |
| Elementary independence statements | p. 15 |
| Problem of axiomatic characterization | p. 16 |
| Classes of probability measures | p. 17 |
| Marginally continuous measures | p. 19 |
| Factorizable measures | p. 22 |
| Multiinformation and conditional product | p. 24 |
| Properties of multiinformation function | p. 27 |
| Positive measures | p. 29 |
| Gaussian measures | p. 30 |
| Basic construction | p. 36 |
| Imsets | p. 39 |
| Graphical Methods | p. 43 |
| Undirected graphs | p. 43 |
| Acyclic directed graphs | p. 46 |
| Classic chain graphs | p. 52 |
| Within classic graphical models | p. 54 |
| Decomposable models | p. 55 |
| Recursive causal graphs | p. 56 |
| Lattice conditional independence models | p. 56 |
| Bubble graphs | p. 57 |
| Advanced graphical models | p. 57 |
| General directed graphs | p. 57 |
| Reciprocal graphs | p. 58 |
| Joint-response chain graphs | p. 58 |
| Covariance graphs | p. 59 |
| Alternative chain graphs | p. 60 |
| Annotated graphs | p. 60 |
| Hidden variables | p. 61 |
| Ancestral graphs | p. 62 |
| MC graphs | p. 62 |
| Incompleteness of graphical approaches | p. 63 |
| Structural Imsets: Fundamentals | p. 65 |
| Basic class of distributions | p. 65 |
| Discrete measures | p. 65 |
| Regular Gaussian measures | p. 66 |
| Conditional Gaussian measures | p. 66 |
| Classes of structural imsets | p. 69 |
| Elementary imsets | p. 69 |
| Semi-elementary and combinatorial imsets | p. 71 |
| Structural imsets | p. 73 |
| Product formula induced by a structural imset | p. 74 |
| Examples of reference systems of measures | p. 75 |
| Topological assumptions | p. 76 |
| Markov condition | p. 78 |
| Semi-graphoid induced by a structural imset | p. 78 |
| Markovian measures | p. 81 |
| Equivalence result | p. 83 |
| Description of Probabilistic Models | p. 87 |
| Supermodular set functions | p. 87 |
| Semi-graphoid produced by a supermodular function | p. 88 |
| Quantitative equivalence of supermodular functions | p. 90 |
| Skeletal supermodular functions | p. 92 |
| Skeleton | p. 93 |
| Significance of skeletal imsets | p. 95 |
| Description of models by structural imsets | p. 99 |
| Galois connection | p. 102 |
| Formal concept analysis | p. 102 |
| Lattice of structural models | p. 104 |
| Equivalence and Implication | p. 111 |
| Two concepts of equivalence | p. 111 |
| Independence and Markov equivalence | p. 113 |
| Independence implication | p. 114 |
| Direct characterization of independence implication | p. 115 |
| Skeletal characterization of independence implication | p. 118 |
| Testing independence implication | p. 120 |
| Testing structural imsets | p. 120 |
| Grade | p. 122 |
| Invariants of independence equivalence | p. 124 |
| Adaptation to a distribution framework | p. 126 |
| The Problem of Representative Choice | p. 131 |
| Baricentral imsets | p. 131 |
| Standard imsets | p. 135 |
| Translation of DAG models | p. 135 |
| Translation of decomposable models | p. 137 |
| Imsets of the smallest degree | p. 141 |
| Decomposition implication | p. 142 |
| Minimal generators | p. 142 |
| Span | p. 145 |
| Determining and unimarginal classes | p. 145 |
| Imsets with the least lower class | p. 146 |
| Exclusivity of standard imsets | p. 148 |
| Dual description | p. 149 |
| Coportraits | p. 149 |
| Dual baricentral imsets and global view | p. 152 |
| Learning | p. 155 |
| Two approaches to learning | p. 155 |
| Quality criteria | p. 161 |
| Criteria for learning DAG models | p. 163 |
| Score equivalent criteria | p. 169 |
| Decomposable criteria | p. 170 |
| Regular criteria | p. 171 |
| Inclusion neighborhood | p. 177 |
| Standard imsets and learning | p. 181 |
| Inclusion neighborhood characterization | p. 181 |
| Regular criteria and standard imsets | p. 184 |
| Open Problems | p. 189 |
| Theoretical problems | p. 189 |
| Miscellaneous topics | p. 189 |
| Classification of skeletal imsets | p. 195 |
| Operations with structural models | p. 199 |
| Reductive operations | p. 199 |
| Expansive operations | p. 202 |
| Cumulative operations | p. 203 |
| Decomposition of structural models | p. 203 |
| Implementation tasks | p. 207 |
| Interpretation and learning tasks | p. 209 |
| Meaningful description of structural models | p. 209 |
| Tasks concerning distribution frameworks | p. 210 |
| Learning tasks | p. 211 |
| Appendix | p. 215 |
| Classes of sets | p. 215 |
| Posets and lattices | p. 216 |
| Graphs | p. 218 |
| Topological concepts | p. 221 |
| Finite-dimensional subspaces and convex cones | p. 222 |
| Linear subspaces | p. 222 |
| Convex sets and cones | p. 223 |
| Measure-theoretical concepts | p. 226 |
| Measure and integral | p. 227 |
| Basic measure-theoretical results | p. 228 |
| Information-theoretical concepts | p. 230 |
| Conditional probability | p. 232 |
| Conditional independence in terms of -algebras | p. 234 |
| Concepts from multivariate analysis | p. 236 |
| Matrices | p. 236 |
| Statistical characteristics of probability measures | p. 238 |
| Multivariate Gaussian distributions | p. 239 |
| Elementary statistical concepts | p. 241 |
| Empirical concepts | p. 242 |
| Statistical conception | p. 244 |
| Likelihood function | p. 245 |
| Testing statistical hypotheses | p. 246 |
| Distribution framework | p. 248 |
| List of Notation | p. 251 |
| List of Lemmas, Propositions etc | p. 259 |
| References | p. 263 |
| Index | p. 273 |
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