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<Paper uid="P02-1046">
  <Title>Bootstrapping</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Problem Setting and Notation
</SectionTitle>
    <Paragraph position="0"> A bootstrapping problem consists of a space of instances X, a set of labels L, a function Computational Linguistics (ACL), Philadelphia, July 2002, pp. 360-367. Proceedings of the 40th Annual Meeting of the Association for Y : X ! L assigning labels to instances, and a space of rules mapping instances to labels. Rules may be partial functions; we write F(x) = ? if F abstains (that is, makes no prediction) on input x. &amp;quot;Classifier&amp;quot; is synonymous with &amp;quot;rule&amp;quot;.</Paragraph>
    <Paragraph position="1"> It is often useful to think of rules and labels as sets of instances. A binary rule F can be thought of as the characteristic function of the set of instances fx : F(x) = +g. Multi-class rules also define useful sets when a particular target class ' is understood. For any rule F, we write F' for the set of instances fx : F(x) = 'g, or (ambiguously) for that set's characteristic function.</Paragraph>
    <Paragraph position="2"> We write -F' for the complement of F', either as a set or characteristic function. Note that -F' contains instances on which F abstains. We write F-' for fx : F(x) 6= '^F(x) 6= ?g. When F does not abstain, -F' and F-' are identical.</Paragraph>
    <Paragraph position="3"> Finally, in expressions like Pr[F = +jY = +] (with square brackets and &amp;quot;Pr&amp;quot;), the functions F(x) and Y(x) are used as random variables.</Paragraph>
    <Paragraph position="4"> By contrast, in the expression P(FjY) (with parentheses and &amp;quot;P&amp;quot;), F is the set of instances for which F(x) = +, and Y is the set of instances for which Y(x) = +.</Paragraph>
  </Section>
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