Composition of Probability Measures on Finite Spaces

Decomposable models and Bayesian net­ works can be defined as sequences of oligo­ dimensional probability measures connected with opemtors of composition. The prelim­ inary results suggest that the probabilistic models allowing for effective computational procedures are represented by sequences pos­ sessing a special property; we shall call them perfect sequences. The present paper lays down the elementary foundation necessary for further study of it­ erative application of operators of composi­ tion. We believe to develop a technique de­ scribing several graph models in a unifying way. We are convinced that practically all theoretical results and procedures connected with decomposable models and Bayesian net­ works can be translated into the terminology introduced in this paper. For example, com­ plexity of computational procedures in these models is closely dependent on possibility to change the ordering of oligo-dimensional mea­ sures defining the model. Therefore, in this paper, lot of attention is paid to possibility to change ordering of the operators of composi­ tion.