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Let ␪ represent the unknown probability of some event occurring. Statistical induction is the task of estimating ␪ from a sequence of observations of the event that it describes. CPT induction is mapped to the statistical induction problem by taking each cell in a CPT to be an unknown conditional probability ␪i, j,k (the cell for node Xi, value xij, and parent instantiation ȏik), and to treat the database as a set of observations S 268 BELIEF MAINTENANCE Table 3. Conditional Probability Tables for the Smoker Network Showing Unknowns That Must Be Learned Pr(P) P ξ (θi, j,k s) = 0 ␪19 1 ␪20 2 ␪21 Pr(S͉P) P S 0 1 2 0 ␪22 ␪23 ␪24 1 ␪25 ␪26 ␪27 Pr(D͉S) S D found as 0 1 b ␪28 ␪29 h ␪30 ␪31 n ␪32 ␪33 of these probabilities.

As with logic, the knowledge is represented declaratively, and so is 265 separated from the inference system. The primary advantages of graphical probabilistic models is that they are perhaps some of the most natural and computationally feasible ways devised yet for managing uncertainty. The representation is visually appealing, the inference mechanisms have a solid statistical and probabilistic foundation, and the approach is a very flexible method for representing beliefs about what factors influence others, and to what extent.

The storage space needed to represent a probability distribution over multiple variables grows exponentially with the number of variables. Concurrently, that implies that the inference in this space would be terribly slow. In the late 1980s, however, the case for graphical probability models as a basis for representing and reasoning about uncertainty was well made by Pearl (3). Part of the argument was that one could take advantage of conditional independence to greatly reduce both the space needed to represent the distribution, and the expected time needed to reason within it.

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