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Up to this point we have estimated probabilities by the fraction of times the event is observed to occur over the total number of opportunities For example, in the above case we estimated P(Wind = strong]Play Tennis = no) by the fraction % where n = 5 is the total number of training examples for which PlayTennis = no, and n, = 3 is the number of these for which Wind = strong While this observed fraction provides a good estimate of the probability in many cases, it provides poor estimates when n, is very small To see the difficulty, imagine that, in fact, the value of P(Wind = strongl PlayTennis = no) is 08 and that we have a sample containing only 5 examples for which PlayTennis = no Then the most probable value for n, is 0 This raises two difficulties First, $ produces a biased underestimate of the probability Second, when this probability estimate is zero, this probability term will dominate the Bayes classifier if the future query contains Wind = strong The reason is that the quantity calculated in Equation (620) requires multiplying all the other probability terms by this zero value To avoid this difficulty we can adopt a Bayesian approach to estimating the probability, using the m-estimate defined as follows

m-estimate of probability:

The first bold section shows the file name used to store your data. You can actually give this any name you like: a.b, for example. The second bold section is where you choose the storage format. Your choices are NSXMLStoreType (the default, as an XML file), NSSQLiteStoreType (for a SQLite database) or NSBinaryStoreType (for the more traditional binary data format used for Mac OS X data stores before the days of Core Data). The Core Data framework is a very large and complex component of Cocoa, and you have barely scratched the surface here. You will return to Core Data programming in much more depth in 9.

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Here, n, and n are defined as before, p is our prior estimate of the probability we wish to determine, and m is a constant called the equivalent sample size, which determines how heavily to weight p relative to the observed data A typical method for choosing p in the absence of other information is to assume uniform

priors; that is, if an attribute has k possible values we set p = For example, in estimating P(Wind = stronglPlayTennis = no) we note the attribute Wind has two possible values, so uniform priors would correspond to choosing p = 5 Note If both n that if m is zero, the m-estimate is equivalent to the simple fraction and m are nonzero, then the observed fraction and prior p will be combined according to the weight m The reason m is called the equivalent sample size is that Equation (622) can be interpreted as augmenting the n actual observations by an additional m virtual samples distributed according to p

Screen savers in Mac OS X, like in other operating systems, are normal applications, but with some additional features that allow them to fit into the screen saver mechanism that is managed within System Preferences In this section, you are going to build a screen saver that makes use of the OpenGL drawing framework I make no pretence of being an OpenGL developer, by the way The example that you will follow here draws on some great resources, notably the chapter on OpenGL programming in Aaron Hillegass s master work Cocoa Programming for Mac OS X, 3rd Edition, and also the excellent articles by Brian Christensen on the Cocoa Dev Central website (wwwcocoadevcentralcom) The code here should be easy to follow, but you can get more detail on the underlying code in these and other resources.

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