A Framework for Bayesian Network Mapping

This research is motivated by the need t o support inference across multiple intelligence systems involving uncertainty. Our objective is to develop a theoretical framework and related inference methods to map semantically similar variables between separate Bayesian networks in a principled way. T he work is to be conducted in two steps. In the first step, we investigate the problem of formalizing the mapping between variables in two separate BNs with different semantics and distributions as pair-wise linkages. In the second step, we aim to justify the mapping between networks as a set of selected variable linkages, and then conduct inference along it. At present, a Bayesian network (BN) is used primarily as a standalone system . When the problem scope is large, a large network slows down inference process and is difficult to review or revise. When the problem itself is distributed, domain knowledge and evidence has to be centralized and unified before a single BN can be created for the problem. Alternatively, separate BNs describing related subdomains or different aspects of the same domain may be created, but it is difficult to combine them for problem solving –– even if the interdependency relations are available. This issue has been investigated in several works, including most notably Multiply Sectioned Bayesian Network (MSBN) by Xiang (Xiang 2002) and Agent Encapsulated Bayesian Network (AEBN) by Valtorta et al. (Valtorta et al, 2002). However, their results are still restricted in scalability, consistency and expressiveness. MSBN’s pair-wise variable linkages are between identical variables with the same distributions, and, to ensure consistency, only one side of the linkage has a complete CPT. AEBN also requires a connection between identical variables, but allows these variables with different distributions. Here, identical variables are the same variables deployed into different BNs. In this paper, we propose a framework that supports inference across BNs through mappings between semantically similar variables. Formalization of BN mapping We modeled BN mapping as a set of four-layered concepts. The first layer is called pair-wise probabilistic relations, which use joint probabilities to represent the dependency between the two variables, which have similar but not necessarily identical semantics and are in two BN. In this framework we assume these joint probabilities are available. Then pair-wise variable linkages, the second layer concept, are created from these probabilistic relations to provide channels for propagating probabilistic influences between the variables across the two BN. The third layer is called valid BN mapping, a selected subset of all available linkages that ensures the consistency of mapped ne tworks. The fourth layer, Minimum valid BN mapping, is obtained by mapping reduction , a process that min imizes the set of linkages while maintain ing the consistency. Figure 1. A Variable Linkage A variable linkage start s from one variable (source variable) and ends at another variable (destination variable) in a different BN. The purpose of building lin kages between variables in different Bayesian networks is to propagate the probability influences from one network to the other. Suppose variable A in BNA and variable B in BNB represent two identical concepts. An o bservation of A (and hence B since A and B are identical) is made in BNA as P (A). This observed distribution of variable B can then be used as soft evidence (denoted se) to update the distributions of BNB (see Valtorta, Kim, and Vomlel 2002) using P(B|se) = P(A). All other variables VB in BNB are then updated by Jeffery’s rule (Pearl 1990):