TEGERA, The International Group of e-Systems Research and Applications. Hammamet, Tunisia, February 12-14, (2007). 1 Loopy Belief Propagation in Bayesian Networks: Origin and possibilistic perspectives Amen Ajroud 1 , Mohamed Nazih Omri 2 , Habib Youssef 3 , Salem Benferhat 4 1 ISET Sousse, TUNISIA - amen.ajroud@isetso.rnu.tn 2 IPEIM Monastir - University of Monastir, TUNISIA - nazih.omri@ipeim.rnu.tn 3 ISITC Hammam-Sousse - University of Monastir, TUNISIA - habib.youssef@fsm.rnu.tn 4 CRIL, Lens - University of Artois, FRANCE - benferhat@cril.univ-artois.fr Abstract : In this paper we present a synthesis of the work performed on two inference algorithms: the Pearl’s belief propagation (BP) algorithm applied to Bayesian networks without loops (i.e. polytree) and the Loopy belief propagation (LBP) algorithm (inspired from the BP) which is applied to networks containing undirected cycles. It is known that the BP algorithm, applied to Bayesian networks with loops, gives incorrect numerical results i.e. incorrect posterior probabilities. Murphy and al. [7] find that the LBP algorithm converges on several networks and when this occurs, LBP gives a good approximation of the exact posterior probabilities. However this algorithm presents an oscillatory behaviour when it is applied to QMR (Quick Medical Reference) network [15]. This phenomenon prevents the LBP algorithm from converging towards a good approximation of posterior probabilities. We believe that the translation of the inference computation problem from the probabilistic framework to the possibilistic framework will allow performance improvement of LBP algorithm. We hope that an adaptation of this algorithm to a possibilistic causal network will show an improvement of the convergence of LBP. 1. Review of Bayesian Networks Bayesian networks are powerful tools for modelling causes and effects in a wide variety of domains. They use graphs capturing causality notion between variables, and probability theory to express the causality power. Bayesian networks are very effective for modelling situations where some information is already known and incoming data is uncertain or partially unavailable. These networks also offer consistent semantics for representing causes and effects via an intuitive graphical representation. Because of all these capabilities, Bayesian networks are regarded as systems for uncertain knowledge representation and have a large number of applications with efficient algorithms and have strong theoretical foundations [9],[10],[11] and [12]. Theoretically, a Bayesian network is a directed acyclic graph (DAG) made up of nodes and causal edges. Each node has a probability of having a certain value. Nodes are often binary, though a Bayesian network may have n-ary nodes. Parent and child nodes are defined as follows: a directed edge exists from a parent to a child. Each child node will have a conditional probability table (CPT) based on parental values. There are no directed cycles in the graph, though there may be “loops”, or undirected cycles. An example network is shown in Figure 1, with parents U i sharing a child X. The node X is a child of the U i 's as well as being a parent to the Y i 's.