Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2005-17

December 2005

When to say “Don’t Know”: Confidence in Automatically Generated Hypotheses without the Assumption of an Underlying Distribution

Iain Morrow

We have a set S, a subset of the n-dimensional Boolean space, together with, for each element x of S, the result of some unknown function F applied to x, and a method for generating a hypothesis h about F given S. We present theoretical and experimental results on four possible methods (similarity, convexification, prevalence and Hamming distance) for determing, given n-dimensional Boolean vectors y,z which are not in S, whether we should be more confident that h(y)=F(y), or that h(z)=F(z), or indeed that we should attach the same degree of confidence to both statements. We consider whether it is possible to have an absolute measure of confidence in the statement that h(b)=F(b) for any given n-dimensional Boolean vector b. We introduce a modification of a standard learning algorithm for Boolean functions, which naturally partitions new examples into three categories: 1,0 and don't know.

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