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On Mathematical Description of Probabilistic Conditional Institute of Information Theory and Automation Academy of Sciences of the Czech Republic On mathematical description of probabilistic conditional indep endence structures Milan Studen y Prague May The thesis for the degree Do ctor of Science in the eld mathematical informatics and theoretical cyb ernetics The thesis was written in the Department of DecisionMaking Theory of the Institute of Information Theory and Automation Academy of Sciences of the Czech Republic and was supp orted by the pro jects K and GACR n It is a result of a longterm research p erformed also within the framework of ESF research program Highly Structured Sto chastic Systems for and the program Causal Interpretation and Identication of Conditional Indep endence Structures of the Fields Institute for Research in Mathematical Sciences University of Toronto Canada September November Address of the author Milan Studen y Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Pod vodarenskou vez Prague Czech Republic email studenyutiacascz web page httpwwwutiacasczuse r datastudenystudeny homehtml Contents Introduction Motivation account Goals of the work Structure of the work Basic concepts Conditional indep endence Semigraphoid prop erties Formal indep endence mo dels Semigraphoids Elementary indep endence statements Problem of axiomatic characterization Classes of probability measures Marginally continuous measures Factorizable measures Multiinformation and conditional pro duct Prop erties of multiinformation function Positive measures Gaussian measures Basic construction Imsets Graphical metho ds Undirected graphs Acyclic directed graphs Classic chain graphs Within classic graphical mo dels Decomp osable mo dels Recursive causal graphs Lattice conditional indep endence mo dels Bubble graphs Advanced graphical mo dels General directed graphs Recipro cal graphs Jointresponse chain graphs Covariance graphs Alternative chain graphs Annotated graphs Hidden variables Ancestral graphs MC graphs Incompleteness of graphical approaches Structural imsets fundamentals Basic class of distributions Discrete measures Nondegenerate Gaussian measures Nondegenerate conditional Gaussian measures Classes of structural imsets Elementary imsets Semielementary and combinatorial imsets Structural imsets Pro duct formula induced by a structural imset Examples of reference systems of measures Topological assumptions Markov condition Semigraphoid induced by a structural imset Markovian measures Equivalence result Description of probabilistic mo dels Sup ermo dular set functions Semigraphoid pro duced by a sup ermo dular function Strong equivalence of sup ermo dular functions Skeletal sup ermo dular functions Skeleton Signicance of skeletal imsets Description of mo dels by structural imsets Galois connection Formal concept analysis Lattice of structural mo dels Markov equivalence Two concepts of equivalence Facial and Markov equivalence Facial implication Direct characterization of facial implication Skeletal characterization of facial implication Adaptation to a distribution framework Testing facial implication Testing structural imsets Grade Invariants of facial equivalence The problem of representative choice Baricentral imsets Standard imsets Translation of DAG mo dels Translation of decomp osable mo dels Imsets of the least degree Strong facial implication Minimal generators Width Determining and unimarginal classes Imsets with the least lower class Exclusivity of standard imsets Other ways of representation Pattern Dual description Op en problems Unsolved theoretical problems Miscellaneous topics Classication of skeletal imsets Op erations with structural mo dels Reductive op erations Expansive op erations Accumulative op erations Implementation tasks Interpretation and learning tasks Meaningful description of structural mo dels Distribution frameworks and learning Conclusions App endix Classes of sets Posets and lattices Graphs .
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