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entropy is between 0 and 1 Figure 32 shows the form of the entropy function relative to a boolean classification, as p, varies between 0 and 1 One interpretation of entropy from information theory is that it specifies the minimum number of bits of information needed to encode the classification of an arbitrary member of S (ie, a member of S drawn at random with uniform probability) For example, if p, is 1, the receiver knows the drawn example will be positive, so no message need be sent, and the entropy is zero On the other hand, if pe is 05, one bit is required to indicate whether the drawn example is positive or negative If pe is 08, then a collection of messages can be encoded using on average less than 1 bit per message by assigning shorter codes to collections of positive examples and longer codes to less likely negative examples Thus far we have discussed entropy in the special case where the target classification is boolean More generally, if the target attribute can take on c different values, then the entropy of S relative to this c-wise classification is defined as

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Entropy(S)

The next thing you need to do is add some meta tags to the <head> element of the welcome page: these help to identify that page to the Help Indexer and Xcode as the entry point for help. See Listing 13 1 much of the <head> element has been omitted for clarity, indicated by [...].

where pi is the proportion of S belonging to class i Note the logarithm is still base 2 because entropy is a measure of the expected encoding length measured in bits Note also that if the target attribute can take on c possible values, the entropy can be as large as log, c

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Given entropy as a measure of the impurity in a collection of training examples, we can now define a measure of the effectiveness of an attribute in classifying the training data The measure we will use, called information gain, is simply the expected reduction in entropy caused by partitioning the examples according to this attribute More precisely, the information gain, Gain(S, A) of an attribute A,

Listing 13 1. Meta Tags in Index.html <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> [...] <meta name="AppleTitle" content="DailyJournal Help" /> <meta name="AppleIcon" content="DailyJournal/DailyJournalIcon16.png" /> <meta name="description" content="Online help for the Consultant's Daily Journal application" /> [...] </head> [...]

Gain(S, A) I Entropy(S) ISVl -Entropy veValues(A) IS1 (S,) (34)

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where Values(A) is the set of all possible values for attribute A, and S, is the subset of S for which attribute A has value v (ie, S, = { s E SIA(s) = v)) Note the first term in Equation (34) is just the entropy of the original collection S, and the second term is the expected value of the entropy after S is partitioned using attribute A The expected entropy described by this second term is simply the sum of the entropies of each subset S,, weighted by the fraction of examples that belong to S, Gain(S, A) is therefore the expected reduction in entropy caused by knowing the value of attribute A Put another way, Gain(S, A) is the information provided about the target &action value, given the value of some other attribute A The value of Gain(S, A) is the number of bits saved when encoding the target value of an arbitrary member of S, by knowing the value of attribute A For example, suppose S is a collection of training-example days described by attributes including Wind, which can have the values Weak or Strong As before, assume S is a collection containing 14 examples, [9+, 5-1 Of these 14 examples, suppose 6 of the positive and 2 of the negative examples have Wind = Weak, and the remainder have Wind = Strong The information gain due to sorting the original 14 examples by the attribute Wind may then be calculated as

The AppTitle meta tag is used, as you will see shortly, to identify the entry point into our help pages. The 16-pixel PNG image will be used in the Help menu of course, there has to be an image file at this location. The description tag provides text that will appear as a tooltip when doing help searches. For this reason you should add meta name="description" tags to each of the HTML files in the collection of help web pages.

Gain(S, Wind) = Entropy(S) -

IS, l -Entropy(S,)

Once you have created and configured the welcome page and other pages, it s time to run the Help Indexer. To do this, start up the program in /Developer/Applications/Utilities/. The interface is very simple, asking for a help folder location. The folder in this case is the folder that contains the index.html and other files (see Figure 13 16):

v~(Weak,Strong]

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The Help Indexer acts in much the same way as a search spider on a conventional web page: it reads the textual content of each web help page and follows any links that it

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