By Uffe B. Kjærulff, Anders L. Madsen

*Bayesian Networks and impact Diagrams: A advisor to building and research, moment Edition,* provides a complete advisor for practitioners who desire to comprehend, build, and learn clever structures for determination aid in line with probabilistic networks. This new version comprises six new sections, as well as fully-updated examples, tables, figures, and a revised appendix. meant essentially for practitioners, this ebook doesn't require refined mathematical abilities or deep knowing of the underlying thought and techniques nor does it speak about replacement applied sciences for reasoning below uncertainty. the speculation and techniques provided are illustrated via greater than one hundred forty examples, and routines are integrated for the reader to ascertain his or her point of knowing. The thoughts and techniques awarded for wisdom elicitation, version development and verification, modeling ideas and methods, studying versions from information, and analyses of versions have all been constructed and subtle at the foundation of various classes that the authors have held for practitioners around the globe.

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4) where X is a subset of the set of variables X. In particular, η(P(X)) = η∅ (P(X)) = P(X), whenever P(X) is a probability distribution over X. 5) X where 1Y denotes a vector of 1s over dom(Y). g. 1Y , is called a vacuous potential. Intuitively, a vacuous potential can be thought of as a non-informative (or superﬂuous) potential. We shall be using the notion of potentials extensively in Chapters 4 and 5, but for now we will just give a couple of simple examples illustrating the usefulness of this notion.

Assume we have an urn with 2 red, 3 green, and 5 blue balls. 5. 8. Similarly, the probability of picking either a red, a green, or a blue is 1. Without replacement, the color of the second ball depends on the color of the ﬁrst ball. If we ﬁrst pick a red ball (and keep it), then the probabilities of picking a red, a green, or a blue ball as the next one are, respectively, P(2nd-is-red | 1st-was-red) = 1 2−1 = , 10 − 1 9 3 3 = , 10 − 1 9 5 5 P(2nd-is-blue | 1st-was-red) = = . 3 on the preceding page we get the probability that the 1st ball is red and the 2nd is red: P(2nd-is-green | 1st-was-red) = P(1st-was-red, 2nd-is-red) = = P(2nd-is-red | 1st-was-red)P(1st-was-red) 1 1 1 · = 9 5 45 Similarly, the probabilities that the 1st ball is red and the 2nd is green/blue are P(1st-was-red, 2nd-is-green) = = P(1st-was-red, 2nd-is-blue) = = P(2nd-is-green | 1st-was-red)P(1st-was-red) 1 1 1 · = , 3 5 15 P(2nd-is-blue | 1st-was-red)P(1st-was-red) 5 1 1 · = , 9 5 9 respectively.

18 2 Networks Probabilistic networks can be seen as compact representations of “fuzzy” cause–eﬀect rules that, contrary to ordinary (logical) rule-based systems, is capable of performing deductive and abductive reasoning as well as inter-causal reasoning. , knowing that a patient suﬀers from angina we can conclude (with high probability) the patient has fever and a sore throat. , observing that a patient has a sore throat provides supporting evidence for angina being the correct diagnosis. The property, however, that sets inference in probabilistic networks apart from other automatic reasoning paradigms is its ability to make inter-causal reasoning: Getting evidence that supports solely a single hypothesis (or a subset of hypotheses) automatically leads to decreasing belief in the unsupported, competing hypotheses.