In many machine learning domains, misclassification costs are different for different examples, in the same way that class membership probabilities are example-dependent. In these domains, both costs and probabilities are unknown for test examples, so both cost estimators and probability estimators must be learned. This paper first discusses how to make optimal decisions given cost and probability estimates, and then presents decision tree learning methods for obtaining well-calibrated probability estimates. The paper then explains how to obtain unbiased estimators for example- dependent costs, taking into account the difficulty that in general, probabilities and costs are not independent random variables, and the training examples for which costs are known are not representative of all examples. The latter problem is called sample selection bias in econometrics. Our solution to it is based on Nobel prize-winning work due to the economist James Heckman. We show that the methods we propose are successful in a comprehensive comparison with MetaCost that uses the well-known and difficult dataset from the KDD'98 data mining contest.
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