By Finn V. Jensen (auth.)

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By the fundamental rule, we have P(U) = P(A I U \ {A})· P(U \ {A}). 16), we get P(U) = P(A I U \ {A})P(U \ {A}) = P(A I pa(A)) . P(U \ {A}). The right-hand side in the preceding formula is the product of all speci0 fied probabilities. 4-4 Bayesian networks admit d-separation In the proof of the chain rule for Bayesian networks, we used d-separation properties induced by the causal network. 1, dseparation is a property of human reasoning and the chain rule is a result of this claim. 8), we can ask whether the d-separation properties are also buried in this formula.

L(H I ft,···, In), where /-L is a normalization constant. 9). 8. A simple Bayes model. The model assumes that the information variables are independent given the hypothesis variable. As can be seen from the insemination example, the assumption need not hold, and if the model is used anyway, the conclusions may be misleading. 5 A simplified poker game In this poker game, each player receives three cards and is allowed two rounds of changing cards. In the first round, you may discard any number of cards from your hand and get replacements from the pack of cards.

33). 6. 6. P(Child I Father, Mother) with aa removed. In order to deal with Fred and Gwenn, we introduce the two unknown fathers, I and K, as mediating variables and assume that they are not sick. For the horses at the top of the network, we specify prior probabilities. This will be an estimate of the frequency of the unwanted gene, and there is no theoretical way to derive it. 01. 0001. Note that the sensitivity to the prior beliefs is very small for the horses where the posterior probability for carrier is well beyond 0, for instance in the cases of Ann and Brian.