M. Studeny: On mathematical description of probabilistic conditional independence structures, thesis for DrSc degree, Institute of Information Thoery and Automation, Academy of Sciences of the Czech Republic, Prague, May 2001, 192 pages.

The first chapter of the thesis is a motivation account. The second chapter is an overview of basic definitions, tools and results concerning the concept of probabilistic conditional independence (CI). The third chapter is an overview of graphical methods for description of probabilistic CI structures which use graphs whose nodes correspond to single random variables. The emphasis is put on classic approaches which use undirected graphs, acyclic directed graphs and chain graphs but advanced graphical methods developed in last six years of 20-th century are mentioned as well. Since graphical methods cannot solve satisfactorily an important theoretical question of completeness a non-graphical method of description of probabilistic CI structures which solves this problem is proposed in the remaining chapters. The method uses certain integer-valued functions on the power set of the set of variables called structural imsets and is safely applicable to the description of CI structures induced by probability measures with finite multiinformation, in particular structures induced by discrete measures, by non-degenerate Gaussian measures and by conditional Gaussian measures. The notion of structural imset is introduced in the fourth chapter where it is shown that three ways of relating structural imsets to probability measures are equivalent. In particular, Markov condition is equivalent to a certain product formula. The fifth chapter introduces another way of description of probabilistic CI structures, namely by means of supermodular functions. It is shown that the class of structural models, that is models which can be described by structural imsets, coincides with the class of models which can be described by supermodular functions. The relation of both ways of description of structural models is lucidly interpreted as a duality relation in terms of an algebraic concept of Galois connection. Atoms and coatoms of the lattice of structural models are characterized and the lattice is shown to be both atomistic and coatomistic finite lattice. The sixth chapter is devoted to relevant implication between structural imsets called facial implication. Two characterizations of facial implication are given and some theoretical questions related to computer implementation of facial implication are solved. Moreover, a method of adaptation of the proposed method to a specific distribution framework is outlined. The seventh chapter deals with the problem of choice of a suitable representative of a class of (facially) equivalent structural imsets and gives some formulas for 'translation' of basic graphical models into structural imsets. The eighth chapter is an overview of open problems to be studied in order to tackle with practical questions. The Appendix is an overview of concepts and facts which are supposed to be elementary and can be omitted by an advanced reader.

AMS classification 62H05, 60A05, 68T30

(probabilistic) conditional independence structures
graphical models
multinformation function
structural imsets
supermodular functions
facial implication
facial and Markov equivalence

A pdf copy (converted postscript version) (1439kB) is available. WARNING: the file is very long, it has 192 pages and contains some colorful pictures.

The thesis presents in a compact updated form a mathematical method developed in the following earlier papers: