Marginalization in Composed Probabilistic Models

Composition of low-dimensional distribu­ tions, whose foundations were laid in the pa­ per published in the Proceedings of UAI'97 (Jirousek 1997), appeared to be an alterna­ tive apparatus to describe multidimensional probabilistic models. In contrast to Graphi­ cal Markov Models, which define multidimen­ sional distributions in a declarative way, this approach is rather procedural. Ordering of low-dimensional distributions into a proper sequence fully defines the respective compu­ tational procedure; therefore, a study of dif­ ferent types of generating sequences is one of the central problems in this field. Thus, it ap­ pears that an important role is played by spe­ cial sequences that are called perfect. Their main characterization theorems are presented in this paper. However, the main result of this paper is a solution to the problem of marginalization for general sequences. The main theorem describes a way to obtain a generating sequence that defines the model corresponding to the marginal of the distri­ bution defined by an arbitrary generating se­ quence. From this theorem the reader can see to what extent these computations are lo­ cal; i.e., the sequence consists of marginal dis­ tributions whose computation must be made by summing up over the values of the vari­ able eliminated (the paper deals with a finite model) .