Composition of Probability Measures on Finite Spaces
This work provides a foundational framework for unifying graph models like decomposable models and Bayesian networks, which could impact probabilistic modeling and computational efficiency in machine learning, though it appears incremental as it builds on existing concepts.
The paper tackles the problem of representing decomposable models and Bayesian networks as sequences of oligo-dimensional probability measures connected by composition operators, identifying 'perfect sequences' as key for enabling effective computational procedures. The result is a foundational framework that unifies graph models and allows translation of theoretical results and procedures, with complexity dependent on ordering changes of these measures.
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.