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where IVSH,DIis the number of hypotheses from H consistent with D The evolution of probabilities associated with hypotheses is depicted schematically in ) Figure 61 Initially (Figure 6 1 ~ all hypotheses have the same probability As training data accumulates (Figures 61 b and 6lc), the posterior probability for inconsistent hypotheses becomes zero while the total probability summing to one is shared equally among the remaining consistent hypotheses The above analysis implies that under our choice for P(h) and P(Dlh),every consistent hypothesis has posterior probability (1/ I V SH, I), and every inconsistent hypothesis has posterior probability 0 Every consistent hypothesis is, therefore, a MAP hypothesis

SARAH RUTH EISENBAUM1 PO Box 92, Brandeis Station Waltham, MA 02254 617-555-8677 7 Evergreen Court Highland Park, IL 60031 708-555-9081

The above analysis shows that in the given setting, every hypothesis consistent with D is a MAP hypothesis This statement translates directly into an interesting statement about a general class of learners that we might call consistent learners We will say that a learning algorithm is a consistent learner provided it outputs a hypothesis that commits zero errors over the training examples Given the above analysis, we can conclude that every consistent learner outputs a MAP hypothesis, i f we assume a uniform prior probability distribution over H (ie, P(hi) = P(hj) for all i, j), and ifwe assume deterministic, noisefree training data (ie, P(D Ih) = 1 i f D and h are consistent, and 0 otherwise) Consider, for example, the concept learning algorithm FIND-Sdiscussed in 2 FIND-Ssearches the hypothesis space H from specific to general hypotheses, outputting a maximally specific consistent hypothesis (ie, a maximally specific member of the version space) Because FIND-Soutputs a consistent hypothesis, we know that it will output a MAP hypothesis under the probability distributions P(h) and P(D1h) defined above Of course FIND-Sdoes not explicitly manipulate probabilities at all-it simply outputs a maximally specific member

In effect, all Cocoa objects inherit the features of NSObjects, though they may also have other features specific to their own class Object orientation is a huge topic and beyond the scope of this book Still, the chances are that you will already have been exposed to the vocabulary of objects, classes, inheritance, methods, and properties If not, there are many books and internet resources that will explain object-orientation..

hypotheses (a)

hypotheses (c)

This is probably only of academic interest to you, but you may like to know where the frameworks are kept on your Mac and how they fit into your Mac environment. The main Apple-supplied frameworks are in /System/Library/Frameworks. Look in this folder and you will see folders for Foundation.framework, AppKit.framework, CoreData.framework, and many others. Let s take a quick look at NSObject. Bear in mind that you won t see the source code for NSObject, or for that matter the compiled framework, but simply the header file. Find and open Foundation.framework, then open the alias Headers. You will see all of the classes within the Foundation framework, and, if you want to know the properties and methods

FIGURE 61 Evolution of posterior probabilities P(hlD) with increasing training data ( a ) Uniform priors assign equal probability to each hypothesis As training data increases first to Dl (b), then to D l A 0 2 (c), the posterior probability of inconsistent hypotheses becomes zero, while posterior probabilities increase for hypotheses remaining in the version space

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of one of these, open it up in a text editor (don t be tempted to try to change it, of course). This is not the best way to find the properties and methods of a Cocoa class, naturally. There is excellent documentation (which you will be exploring in 8). The point of this exercise is simply to familiarize yourself with the frameworks and dispel some of the mystery that may surround them.