31.Knowledge and Data Engineering by John G. Webster (Editor)

By John G. Webster (Editor)

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The density function f(Bp͉Bs) is uniform, where Bp is the quantitative structure, or the probabilities. This says that there is no initial preference as to what probabilities to place on the structure Bs. Assumptions 1 and 3 can be relaxed by dealing probabilistically with hidden variables and missing values [see Cooper (13) and York and Madigan (14)]. The last assumption can be relaxed to allow a Dirichlet prior. Finally, from Bayes rule, it is clear that the probability of a structure given a database of examples D is found as Pr(Bs͉D) ϭ Pr(Bs, D)/Pr(D).

Knowledge in a belief network is represented in the form of conditional probabilities. 75. Inference in the belief network involves manipulating the conditional probabilities in ways consistent with Bayes rule and the axioms of probability. Belief networks can be thought of as representing sequences of causal events. This orientation is helpful for constructing the networks, although it is not mathematically necessary. From the smoking example, one could imagine that (1) parents that smoke cause their children to smoke, and (2) that causality must be represented in the form used previously.

The distinction is important in guiding the types of assumptions that can be made in the algorithms used to construct a system of beliefs on the basis of empirical evidence. For a clear explanation of the taxonomy of semantic interpretations of probability, refer to Walley (4). In this article, probabilities are epistemic unless otherwise specified. The rest of this article will focus on one particular type of graphical probability model: a belief network. The technical challenges, issues, and limitations of this approach which are discussed are largely shared with other graphical probability models.

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