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By Finn V. Jensen (auth.)

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To achieve P(A, e), we marginalize the variables B, C, G, and H out of P(A, B, C, d, j, G, H). 5), so let us start with G; that is, we wish to calculate LP(A,B,C,d,j,G,H) = G LP(A)P(H)P(B I A,H)P(C I A)P(d I B,H)P(f I B,C)P(G I C). 5) we have 'LP(A,B,C,d,j,G,H) = G P(A)P(H)P(B I A,H)P(C I A)P(d I B,H)P(f I B,C) 'LP(G I C), G and we need only calculate L,G P(G I C). Actually, for each state c of C, we have L,G P(G I c) = 1, and we need not perform any calculation at all. We have P(A, B, C, d, j, H) = L P(A, B, C, d, j, G, H) = G P(A)P(H)P(B I A,H)P(C I A)P(d I B,H)P(f I B,C).

Therefore, we require that the network does not contain cycles. Definition A Bayesian network consists of the following: A set of variables and a set of directed edges between variables. Each variable has a finite set of mutually exclusive states. The variables together with the directed edges form a directed acyclic graph (DAG). (A directed graph is acyclic if there is no directed path Al -+ ... -+ An s. t. ) To each variable A with parents BI, ... , B n , there is attached the potential table P(A I B I ,···, Bn).

Instead, we require that the d-separation properties implied by the structure hold. 20 1. 14. A directed acyclic graph (DAG). The probabilities to specify are P(A), P(B), P(C I A,B), P(E I C), P(D I C), P(F I E), and P(G I D,E,F). 4). When building thestructure of Bayesian network models, we need not insist on having the links go in a causal direction. On the other hand, we then need to check its d-separation properties to ensure that they correspond with our perception of the world. 15. 15. The causal network for the reduced car start problem.

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