barcode in vb.net 2010 INDUCTIVE-ANALYTICALAPPROACHES TO LEARNING 1221 The Learning Problem in Software

Print Denso QR Bar Code in Software INDUCTIVE-ANALYTICALAPPROACHES TO LEARNING 1221 The Learning Problem

122 INDUCTIVE-ANALYTICALAPPROACHES TO LEARNING 1221 The Learning Problem
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To summarize, the learning problem considered in this chapter is
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Given:
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A set of training examples D, possibly containing errors A domain theory B, possibly containing errors A space of candidate hypotheses H
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Determine:
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A hypothesis that best fits the training examples and domain theory What precisely shall we mean by "the hypothesis that best fits the training examples and domain theory 'In particular, shall we prefer hypotheses that fit
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the data a little better at the expense of fitting the theory less well, or vice versa We can be more precise by defining measures of hypothesis error with respect to the data and with respect to the domain theory, then phrasing the question in terms of these errors Recall from 5 that errorD(h) is defined to be the proportion of examples from D that are misclassified by h Let us define the error e r r o r ~ ( h of h with respect to a domain theory B to be the probability that h ) will disagree with B on the classification of a randomly drawn instance We can attempt to characterize the desired output hypothesis in terms of these errors For example, we could require the hypothesis that minimizes some combined measure of these errors, such as argmin kDerrorD( h )
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+ kBerrorB( h )
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While this appears reasonable at first glance, it is not clear what values to assign to k ~ and kg to specify the relative importance of fitting the data versus fitting the theory If we have a very poor theory and a great deal of reliable data, it will be ) best to weight e r r o r ~ ( hmore heavily Given a strong theory and a small sample of very noisy data, the best results would be obtained by weighting errorB(h) more heavily Of course if the learner does not know in advance the quality of the domain theory or training data, it will be unclear how it should weight these two error components An alternative perspective on the question of how to weight prior knowledge and data is the Bayesian perspective Recall from 6 that Bayes theorem describes how to compute the posterior probability P(h1D) of hypothesis h given observed training data D In particular, Bayes theorem computes this posterior probability based on the observed data D , together with prior knowledge in the form of P ( h ) , P ( D ) , and P(Dlh) Thus we can think of P ( h ) , P ( D ) , and P(Dlh) as a form of background knowledge or domain theory, and we can think of Bayes theorem as a method for weighting this domain theory, together with the observed data D , to assign a posterior probability P(hlD) to h The Bayesian view is that one should simply choose the hypothesis whose posterior probability is greatest, and that Bayes theorem provides the proper method for weighting the contribution of this prior knowledge and observed data Unfortunately, Bayes theorem implicitly assumes pe$ect knowledge about the probability distributions P ( h ) , P ( D ) , and P ( D l h ) When these quantities are only imperfectly known, Bayes theorem alone does not prescribe how to combine them with the observed data (One possible approach in such cases is to assume prior probability distributions over P ( h ) , P ( D ) , and P(D1h) themselves, then calculate the expected value of the posterior P (h1 D ) However, this requires additional knowledge about the priors over P ( h ) , P ( D ) , and P(Dlh), so it does not really solve the general problem) We will revisit the question of what we mean by "best" fit to the hypothesis and data as we examine specific algorithms For now, we will simply say that the learning problem is to minimize some combined measure of the error of the hypothesis over the data and the domain theory
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