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A Tutorial on Bayesian Belief Networks Mark L Krieg Surveillance Systems Division Electronics and Surveillance Research Laboratory DSTO{TN{0403 ABSTRACT This tutorial provides an overview of Bayesian belief networks. The sub- ject is introduced through a discussion on probabilistic models that covers probability language, dependency models, graphical representations of mod- els, and belief networks as a particular representation of probabilistic models. The general class of causal belief networks is presented, and the concept of d-separation and its relationship with independence in probabilistic models is introduced. This leads to a description of Bayesian belief networks as a speciﬂc class of causal belief networks, with detailed discussion on belief propagation and practical network design. The target recognition problem is presented as an example of the application of Bayesian belief networks to a real problem, and the tutorial concludes with a brief summary of Bayesian belief networks. APPROVED FOR PUBLIC RELEASE

DSTO{TN{0403 Published by DSTO Electronics and Surveillance Research Laboratory PO Box 1500 Edinburgh, South Australia, Australia 5111 Telephone: Facsimile: (08) 8259 5555 (08) 8259 6567 c Commonwealth of Australia 2001 AR No. 012{084 December, 2001 APPROVED FOR PUBLIC RELEASE ii

A Tutorial on Bayesian Belief Networks DSTO{TN{0403 EXECUTIVE SUMMARY A Bayesian belief network is a graphical representation of a probabilistic dependency model. It consists of a set of interconnected nodes, where each node represents a variable in the dependency model and the connecting arcs represent the causal relationships between these variables. Each node or variable may take one of a number of possible states or values. The belief in, or certainty of, each of these states is determined from the belief in each possible state of every node directly connected to it and its relationship with each of these nodes. The belief in each state of a node is updated whenever the belief in each state of any directly connected node changes. Bayesian belief networks are particularly suited to the target recognition problem, where the category, identity and class of a target track are to be determined. Each of these three track attributes may be modelled by a hypothesis node, in which each state represents a diﬁerent hypothesis. Evidence, such as Identiﬂcation Friend or Foe (IFF) reports, Electronic Support (ES) data and track dynamics, is applied to the network through evidence nodes. On receipt of evidence, the belief in the state of the evidence node changes, causing changes in the belief of all nodes to ripple through the entire network, including the hypothesis nodes. In this way, the evidence updates the beliefs for each category, identity and class, and possibly the most likely state of each. iii

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DSTO{TN{0403 Author Mark L Krieg Surveillance Systems Division Mark Krieg joined the Defence Science and Technology Organ- isation (DSTO) Australia in 1976 as a radio apprentice. From 1981 until 1987 he worked as a technical o–cer in the Commu- nications and Electronic Engineering Division in the areas of communication networks, and control and instrumentation. Dr Krieg obtained his BE(Elect) from the University of Ade- laide in 1992 and his PhD from the same institution in 1998. He joined the Microwave Radar Division in 1992 where he worked in the radar signal and data processing area. He is currently a Senior Research Scientist attached to the Tracking and Sensor Fusion group of the Surveillance Systems Division, where he is pursuing research into multi-sensor tracking and fusion for defence applications. v

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DSTO{TN{0403 Contents Glossary Abbreviations and Acronyms Symbols 1 Introduction 2 Probabilistic Models 2.1 Probability Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Dependency Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Graphical Representation of Dependency Models . . . . . . . . . . . . . . 2.4 Belief Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Causal Belief Networks 3.1 Dependencies in Causal Networks . . . . . . . . . . . . . . . . . . . . . . 4 Bayesian Belief Networks 5 Propagation in Bayesian Belief Networks xi xiv xiv 1 2 3 4 4 5 6 7 10 12 5.1 Causal Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Causal Polytrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.3 Practical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.3.1 5.3.2 5.3.3 5.3.4 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Stochastic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 19 The Noisy-Or-Gate . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6 Building Networks 22 6.1 Choosing the Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6.2 The Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.3 Other Design Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.3.1 6.3.2 6.3.3 6.3.4 Undirected Relations . . . . . . . . . . . . . . . . . . . . . . . . . 25 Divorcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Noisy Or . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Causal Independence . . . . . . . . . . . . . . . . . . . . . . . . . 26 vii

DSTO{TN{0403 7 Target Recognition Application 8 Summary References Appendices A Axioms for Dependency Models B Markov Networks 26 27 28 30 35 B.1 Markov Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 B.2 Join Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 C Belief Table Algebra 39 C.1 Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 C.2 Marginalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 D Learning 41 D.1 Batch Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 D.2 Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 E Comparison with Decision Theory Figures 44 8 9 9 3.1 3.2 3.3 5.1 5.2 5.3 5.4 5.5 5.6 5.7 A serial belief network connection . . . . . . . . . . . . . . . . . . . . . . . . . A diverging belief network connection . . . . . . . . . . . . . . . . . . . . . . A converging belief network connection . . . . . . . . . . . . . . . . . . . . . . Bayesian belief network for track identiﬂcation from platform data . . . . . . 12 Propagation in a chain network . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Propagation in a tree network . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Propagation in a causal polytree network . . . . . . . . . . . . . . . . . . . . 15 Simple clustering example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Clustering using join trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Using conditioning to remove loops . . . . . . . . . . . . . . . . . . . . . . . . 19 5.8 Modelling disjunctive interactions using the noisy-OR-gate . . . . . . . . . . . 21 viii

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