Radim Jirousek

Radim Jirousek

Decomposable models and Bayesian networks can be defined as sequences of oligo-dimensional probability measures connected with operators of composition. The preliminary results suggest that the probabilistic models allowing for effective computational procedures are represented by sequences possessing a special property; we shall call them perfect sequences. The paper lays down the elementary foundation necessary for further study of iterative application of operators of composition. We believe to develop a technique describing several graph models in a unifying way. We are convinced that practically all theoretical results and procedures connected with decomposable models and Bayesian networks can be translated into the terminology introduced in this paper. For example, complexity of computational procedures in these models is closely dependent on possibility to change the ordering of oligo-dimensional measures defining the model. Therefore, in this paper, lot of attention is paid to possibility to change ordering of the operators of composition.

Radim Jirousek

Composition of low-dimensional distributions, whose foundations were laid in the papaer published in the Proceeding of UAI'97 (Jirousek 1997), appeared to be an alternative apparatus to describe multidimensional probabilistic models. In contrast to Graphical Markov Models, which define multidomensinoal distributions in a declarative way, this approach is rather procedural. Ordering of low-dimensional distributions into a proper sequence fully defines the resepctive computational procedure; therefore, a stury of different type of generating sequences is one fo the central problems in this field. Thus, it appears that an important role is played by special sequences that are called perfect. Their main characterization theorems are presetned in this paper. However, the main result of this paper is a solution to the problem of margnialization for general sequences. The main theorem describes a way to obtain a generating sequence that defines the model corresponding to the marginal of the distribution defined by an arbitrary genearting sequence. From this theorem the reader can see to what extent these comutations are local; i.e., the sequence consists of marginal distributions whose computation must be made by summing up over the values of the variable eliminated (the paper deals with finite model).