interval belief degree estimates (imprecise probabilities)
Algebraical Bayesian networks (ABN) are considered as a mathematical model for knowledge patterns bases with uncertainty. An algebraical Bayesian network is a set of conjuncts ideals with scalar or interval conjunct probability estimates; the set is structures as a join graph.
The global probabilistic-logic inference in an algebraical Bayesian networks acts in the whole set of knowledge patterns belonging to the ABN. The knowledge patterns interconnections structure is sufficiently used in the inference; moreover, this structure influences on computational complexity of the global inference operations as well as on the possibility of these operations per-forming.
Several global probabilistic-logic inference operations in algebraical Bayesian networks are defined: 1) algebraical Bayesian network consistency degree verification and reconciliation; there are four consistency degrees: local, external, internal, global; 2) global à posteriori inference – evidence propagation; there are three types of evidence: deterministic, stochastic and imprecise; 3) building linear combination and linear span over a set of consistent algebraical Bayesian networks as well as the operation of enclosing consistency that results in a consistent algebraical Bayesian network which element probabilistic estimate intervals contain (enclose) corresponding element probabilistic estimate intervals from the initial set of algebraical Bayesian networks.
Several relationships among algebraical Bayesian network consistency degrees are discovered. The most important relationship concerns the fact that in case of acyclic algebraical Bayesian network its interval consistency guarantees its global consistency. This result allows for considerable reduction in computational complexity required for algebraical Bayesian network global consistency verification and reconciliation.
Global à posteriori inference is based on the process of virtual evidence propagation. This process is valid only for acyclic algebraical Bayesian networks. Any knowledge patterns cycle and certain classes of algebraical Bayesian networks with such cycles can be rationally transformed into acyclic algebraical Bayesian networks. This transformation allows for further global à posteriori inference as well as for global consistency verification and reconciliation with less computational complexity. However the transformation under question can be implemented with several means and it opens a wide area for further extended studies.
The state of consistency (according to a specified degree) remains (is reserved) relatively linear combination and linear span operations. The linear span operation yields the minimal (relatively to probabilistic estimate interval hierarchy) algebraical Bayesian network that corresponds to the definition of enclosing consistency related to a set of consistent algebraical Bayesian network sharing the same structure.
An acyclic Bayesian belief network and feedback cycles can be transformed into a corresponding acyclic algebraical Bayesian network and the probabilistic semantics of the initial network or cycle remains unchanged under this transformation.