The law is only proved for discrete valued learning tasks, but there is no reason to believe it does not also apply to continuous valued tasks
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...only
To provide an example of an implausible similarity metric, consider the similarity metric defined by the root node, that everything is similar. This will not be plausible as there is too great a level of dissimilarity in classes with respect to this metric. If it were a relevant similarity metric, and the distribution of training examples was representative of the distribution of objects in the domain as a whole, then the similarity assumption would be violated, as similar objects would have probability of just 0.58 of belonging to the same class. This probability can be calculated as follows. The probabilities of an object being + or - are 0.3 and 0.7 respectively. If an object is + then the probability of it belonging to the same class as another object to which it is similar is 0.3. If an object is - then the probability of it belonging to the same class as another object to which it is similar is 0.7. Thus, the probability of an object belonging to the same class as another similar object is 4#4. The numbers involved in this simple example are, of course, too small to reach any such conclusion with a high level of confidence-the example is intended as illustrative only.
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Geoff Webb
Mon Sep 9 12:13:30 EST 1996