Introduction to Bayesian networks
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1 PhD seminar series Probabilistics in Engineering : Bayesian networks and Bayesian hierarchical analysis in engineering Conducted by Prof. Dr. Maes, Prof. Dr. Faber and Dr. Nishijima Introduction to Bayesian networks Kazuyoshi Nishijima IBK, ETH Zurich 1/20
2 Contents Eidgenossische Technische Hochschule Zürich What is a Bayesian network? Model building Algorithms for inferences Software tools 2/20
3 What is a Bayesian network? Definition (Jensen and Nielsen (2007)) A Bayesian network consists of the following: A set of variables and a set of directed edges between variables. The variables together with the directed edges form an acyclic directed graph (DAG). Each variable has a finite set of mutually exclusive states. To each variable, there is a conditional probability table attached. Jensen, F.V. and Nielsen, T.D. (2007) Bayesian Networks and Decision Graphs second edition, Springer. 3/20
4 Remark The third and fourth properties p may be extended properly p so that a network allows for considering continuous variables. 4/20
5 Example of a Bayesian network 5/20
6 Types of connections Serial connection Diverging connection Converging connection Any connections in Bayesian networks belong to one of the three types. 6/20
7 d-separation Definition (Jensen and Nielsen (2007)) Two distinct variables A and B in a network are d-separated if for all paths between A and B, there is an intermediate variable V (distinct from A and B) such that either the connection is serial or diverging and V is instantiated or the connection is converging, and neither V nor any of V s descendants have received evidence. concerns conditional independency. 7/20
8 Questions Eidgenossische Technische Hochschule Zürich E is d-separated to which? F is d-separated to which? Fig in Jensen and Nielsen (2007). 8/20
9 Causal relation in Bayesian networks Direct arrows in accordance with causal relations! Why? 9/20
10 Causal relation in Bayesian networks Because it leads to a minimal representation of the problem in consideration. The network on the left is not appropriate, because BBQ party and Forecast are independent. 10/20
11 Chain rule for Bayesian networks Joint probability representation as a product of conditional probabilities. where { } P( U) = P( Ai pa( Ai)) U = A1, A2,..., An is the states of all the variables, represents the state of i th variable, A i i pa ( A ) represents the states of the parents of i th i variable. A compact representation of joint probability 11/20
12 Divorcing - a useful technique Intermediate variable Size of the conditional probability tables is reduced. Fig in Jensen and Nielsen (2007). 12/20
13 Extensions of Bayesian networks Object-oriented Bayesian networks Dynamic Bayesian networks 13/20
14 Example of the object-oriented oriented Bayesian networks Nishijima K., Maes M.A., Goyet J., Faber M.H.. Constrained optimization of component reliabilities in complex systems. Structural Safety 2009; /20
15 Example of the object-oriented oriented Bayesian networks 15/20
16 Example of dynamic Bayesian networks Straub, D. (2009) An efficient computational framework for probabilistic deterioration modeling and reliability updating, ICOSSAR2009, Osaka, Japan. 16/20
17 Some algorithms for inferences Exact inferences Junction tree propagation Approximate inferences Loopy belief updating Variational approximations Stochastic inferences Likelihood weighting Gibbs sampling Implementation into Bayesian networks by means of Discretization Mixture of truncated exponentials (MTE) etc. Metropolis-Hasting sampling Slice sampling See Langseth et al. (2009) and Jensen and Nielsen (2007). Langeth, H., Nielsen, T.D., Rumi, R. and Salmeron, A. (2009) Inference in hybrid Bayesian networks, Reliability Engineering and System Safety, 94, pp /20
18 Software for Graphical models/bayesian networks More than 50 software tools are available as of August 2009: see Among others Hugin ( GeNIe ( Winbugs ( ac uk/bugs/) 18/20
19 Demonstration of the use of software tools Hugin GeNIe Winbugs 19/20
20 Administration Please send me your the file for your presentation by one day before the presentation. Duration of the presentation is about minutes. Presented materials are uploaded, if not disagreed. Self-introduction including your interests/applications of Bayesian networks. 20/20
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