M. Studeny:
Description of structures of stochastic conditional independence
by means of faces and imsets. (a series of 3 papers)
1st part:
introduction and basic concepts. International Journal of General
Systems 23 (1994), n. 2, p. 123137.
2nd part:
basic theory. International Journal of General Systems 23
(1995), n. 3, pp. 201219.
3rd part:
examples of use and appendices. International Journal
of General Systems 23 (1995), n. 4, pp. 323341.
 Global abstract (for all three parts)
 The work presents a new approach to the mathematical description of
stochastic conditional independence structures among a finite number
of random variables. The new approach is related to the classic approaches,
that is to the use of directed acyclic graphs (probabilistic influence
diagrams or Bayesian nets), undirected graphs (Markov nets) and
dependency models (namely semigraphoids). The approach provides a deductive
mechanism to infer probabilistically valid consequences of input
affirmative information about conditional independence structure. This
mechanism is much more powerful than the use of semigraphoids as it
includes, from the classic point of view, an infinite number of
inference rules. Nevertheless, from the theoretical point of view, it
is finitely implementable. The developed theory is illustrated by examples
showing how it is computer managable.
The series of papers is in principle selfcontained, but several
propositions have their proofs based on the results of the paper:
 M. Studeny:
Convex cones in finitedimensional real vector spaces.
Kybernetika 29 (1993), n. 2, pp. 180200.
 Abstract of the first part
 The first part contains the Introduction in which the history of the
description of conditional independence structures is recalled and the
construction of the presented theory is explained by means of illustrative
diagrams. Then basic concepts including the concept of imset are
defined or recalled. Finally, several assertions concerning quasiorderings
on imsets are presented.
 Abstract of the second part
 The central concept of face (with respect to an arbitrary continous
linear ordering on imsets) and the corresponding deductive mechanism (facial
implication) are introduced. The class of all faces constitutes a lattice
which, in the case of a finitely established ordering, is shown to be finite.
Moreover, the atoms and coatoms of the lattice are characterized and two
possible representations of faces are emphasized: by means of
generating imsets and by means of portraits. Then a particular
ordering, called the structural ordering, is studied. The class
of faces with respect to this ordering is identified with a certain
class of dependency models which includes all models of probabilistic
conditional independence structures.
 Abstract of the third part
 This part contains several examples that indicate how to implement
the facial deductive mechanism and show how to transform
information about conditional independence structure given in the
form of a dependency model, influence diagram, or Markov net into
the form of an imset. Another example shows that the facial deductive
mechanism is
indeed more powerful than the semigraphoid one. A simple method of proving
the probabilistic soundness of an inference rule is presented. The
advantages of, disadvantages of, and prospects for the presented approach are
discussed in the Conclusions. Two appendices contain some supplementary
results about the structural ordering of imsets.
 AMS classification 62H05, 68T30, 52B99, 94A17
 Keywords
 conditional independence
 imset
 face
 semigraphoid
 portrait
 skeleton
 structural ordering (imset, face)
 A
scanned pdf copy of the first part (925kB) is available.
 A
scanned pdf copy of the second part (1050kB) is available.
 A
scanned pdf copy of the third part (1048kB) is available.

Moreover, the content of the series of papers is covered by the
DrSc thesis "On mathematical description of probabilistic conditional independence
structures".