Introduction / 1: |
Motivational thoughts / 1.1: |
Goals of the monograph / 1.2: |
Structure of the book / 1.3: |
Basic Concepts / 2: |
Conditional independence / 2.1: |
Semi-graphoid properties / 2.2: |
Formal independence models / 2.2.1: |
Semi-graphoids / 2.2.2: |
Elementary independence statements / 2.2.3: |
Problem of axiomatic characterization / 2.2.4: |
Classes of probability measures / 2.3: |
Marginally continuous measures / 2.3.1: |
Factorizable measures / 2.3.2: |
Multiinformation and conditional product / 2.3.3: |
Properties of multiinformation function / 2.3.4: |
Positive measures / 2.3.5: |
Gaussian measures / 2.3.6: |
Basic construction / 2.3.7: |
Imsets / 2.4: |
Graphical Methods / 3: |
Undirected graphs / 3.1: |
Acyclic directed graphs / 3.2: |
Classic chain graphs / 3.3: |
Within classic graphical models / 3.4: |
Decomposable models / 3.4.1: |
Recursive causal graphs / 3.4.2: |
Lattice conditional independence models / 3.4.3: |
Bubble graphs / 3.4.4: |
Advanced graphical models / 3.5: |
General directed graphs / 3.5.1: |
Reciprocal graphs / 3.5.2: |
Joint-response chain graphs / 3.5.3: |
Covariance graphs / 3.5.4: |
Alternative chain graphs / 3.5.5: |
Annotated graphs / 3.5.6: |
Hidden variables / 3.5.7: |
Ancestral graphs / 3.5.8: |
MC graphs / 3.5.9: |
Incompleteness of graphical approaches / 3.6: |
Structural Imsets: Fundamentals / 4: |
Basic class of distributions / 4.1: |
Discrete measures / 4.1.1: |
Regular Gaussian measures / 4.1.2: |
Conditional Gaussian measures / 4.1.3: |
Classes of structural imsets / 4.2: |
Elementary imsets / 4.2.1: |
Semi-elementary and combinatorial imsets / 4.2.2: |
Structural imsets / 4.2.3: |
Product formula induced by a structural imset / 4.3: |
Examples of reference systems of measures / 4.3.1: |
Topological assumptions / 4.3.2: |
Markov condition / 4.4: |
Semi-graphoid induced by a structural imset / 4.4.1: |
Markovian measures / 4.4.2: |
Equivalence result / 4.5: |
Description of Probabilistic Models / 5: |
Supermodular set functions / 5.1: |
Semi-graphoid produced by a supermodular function / 5.1.1: |
Quantitative equivalence of supermodular functions / 5.1.2: |
Skeletal supermodular functions / 5.2: |
Skeleton / 5.2.1: |
Significance of skeletal imsets / 5.2.2: |
Description of models by structural imsets / 5.3: |
Galois connection / 5.4: |
Formal concept analysis / 5.4.1: |
Lattice of structural models / 5.4.2: |
Equivalence and Implication / 6: |
Two concepts of equivalence / 6.1: |
Independence and Markov equivalence / 6.1.1: |
Independence implication / 6.2: |
Direct characterization of independence implication / 6.2.1: |
Skeletal characterization of independence implication / 6.2.2: |
Testing independence implication / 6.3: |
Testing structural imsets / 6.3.1: |
Grade / 6.3.2: |
Invariants of independence equivalence / 6.4: |
Adaptation to a distribution framework / 6.5: |
The Problem of Representative Choice / 7: |
Baricentral imsets / 7.1: |
Standard imsets / 7.2: |
Translation of DAG models / 7.2.1: |
Translation of decomposable models / 7.2.2: |
Imsets of the smallest degree / 7.3: |
Decomposition implication / 7.3.1: |
Minimal generators / 7.3.2: |
Span / 7.4: |
Determining and unimarginal classes / 7.4.1: |
Imsets with the least lower class / 7.4.2: |
Exclusivity of standard imsets / 7.4.3: |
Dual description / 7.5: |
Coportraits / 7.5.1: |
Dual baricentral imsets and global view / 7.5.2: |
Learning / 8: |
Two approaches to learning / 8.1: |
Quality criteria / 8.2: |
Criteria for learning DAG models / 8.2.1: |
Score equivalent criteria / 8.2.2: |
Decomposable criteria / 8.2.3: |
Regular criteria / 8.2.4: |
Inclusion neighborhood / 8.3: |
Standard imsets and learning / 8.4: |
Inclusion neighborhood characterization / 8.4.1: |
Regular criteria and standard imsets / 8.4.2: |
Open Problems / 9: |
Theoretical problems / 9.1: |
Miscellaneous topics / 9.1.1: |
Classification of skeletal imsets / 9.1.2: |
Operations with structural models / 9.2: |
Reductive operations / 9.2.1: |
Expansive operations / 9.2.2: |
Cumulative operations / 9.2.3: |
Decomposition of structural models / 9.2.4: |
Implementation tasks / 9.3: |
Interpretation and learning tasks / 9.4: |
Meaningful description of structural models / 9.4.1: |
Tasks concerning distribution frameworks / 9.4.2: |
Learning tasks / 9.4.3: |
Appendix / A: |
Classes of sets / A.1: |
Posets and lattices / A.2: |
Graphs / A.3: |
Topological concepts / A.4: |
Finite-dimensional subspaces and convex cones / A.5: |
Linear subspaces / A.5.1: |
Convex sets and cones / A.5.2: |
Measure-theoretical concepts / A.6: |
Measure and integral / A.6.1: |
Basic measure-theoretical results / A.6.2: |
Information-theoretical concepts / A.6.3: |
Conditional probability / A.6.4: |
Conditional independence in terms of ?-algebras / A.7: |
Concepts from multivariate analysis / A.8: |
Matrices / A.8.1: |
Statistical characteristics of probability measures / A.8.2: |
Multivariate Gaussian distributions / A.8.3: |
Elementary statistical concepts / A.9: |
Empirical concepts / A.9.1: |
Statistical conception / A.9.2: |
Likelihood function / A.9.3: |
Testing statistical hypotheses / A.9.4: |
Distribution framework / A.9.5: |
List of Notation |
List of Lemmas, Propositions etc |
References |
Index |
Introduction / 1: |
Motivational thoughts / 1.1: |
Goals of the monograph / 1.2: |