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For example, to calculate the derivative of In P(D1h) with respect to the upperrightmost entry in the table of Figure 63 we will have to calculate the quantity P(Campf ire = True, Storm = False, BusTourGroup = Falseld) for each training example d in D When these variables are unobservable for the training example d , this required probability can be calculated from the observed variables in d using standard Bayesian network inference In fact, these required quantities are easily derived from the calculations performed during most Bayesian network inference, so learning can be performed at little additional cost whenever the Bayesian network is used for inference and new evidence is subsequently obtained Below we derive Equation (625) following Russell et al (1995) The remainder of this section may be skipped on a first reading without loss of continuity To simplify notation, in this derivation we will write the abbreviation Ph(D) to represent P ( D J h ) Thus, our problem is to derive the gradient defined by the set of derivatives for all i , j, and k Assuming the training examples d in the data set D are drawn independently, we write this derivative as

The one area of slight inconsistency is the presence of the HUD window (circled in Figure 14 1). This gray, semitransparent window is a recent introduction and is reminiscent of the UI of Apple s professional applications, such as Aperture. Does this perhaps indicate a move to a wider choice of user interface options for the developer Time alone will tell.

= 1ax W can now f(~) This last step makes use of the general equality introduce the values of the variables Yi and Ui = Parents(Yi), by summing over their possible values yijl and uiu

This last step follows from the product rule of probability, Table 61 Now consider the the rightmost sum in the final expression above Given that W i j k = Ph(yijl~ik), is nonzero is the term for which j' = j and only term in this sum for which i' = i Therefore

I ve mentioned it already, but it bears repeating. This book is not about learning to write Objective-C code. The focus is on introducing you to, and making productive use of, the Xcode Tools. Along the way, there will be many completely worked examples that are obviously written in the language, but there will not be extensive discussion of language syntax or structure. There are plenty of great books and other resources out there to help you with that. This book is a learning book, and as such is not aimed at advanced developers. As a result, many of the topics are introduced and described to a sufficient level of detail to let you being to be productive, but no further. If I were to cover every topic in exhaustive detail, you d be holding a book of over 1000 pages right now!

Thus, we have derived the gradient given in Equation (625) There is one more item that must be considered before we can state the gradient ascent training procedure In particular, we require that as the weights wijk are updated they must remain valid probabilities in the interval [0,1] We also require that the sum wijk remains 1 for all i , k These constraints can be satisfied by updating weights in a two-step process First we update each wijkby gradient ascent

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where q is a small constant called the learning rate Second, we renormalize the weights wijk to assure that the above constraints are satisfied As discussed by Russell et al, this process will converge to a locally maximum likelihood hypothesis for the conditional probabilities in the Bayesian network As in other gradient-based approaches, this algorithm is guaranteed only to find some local optimum solution An alternative to gradient ascent is the EM algorithm discussed in Section 612, which also finds locally maximum likelihood solutions

Summary

Learning Bayesian networks when the network structure is not known in advance is also difficult Cooper and Herskovits (1992) present a Bayesian scoring metric for choosing among alternative networks They also present a heuristic search algorithm called K2 for learning network structure when the data is fully observable Like most algorithms for learning the structure of Bayesian networks, K2 performs a greedy search that trades off network complexity for accuracy over the training data In one experiment K2 was given a set of 3,000 training examples generated at random from a manually constructed Bayesian network containing 37 nodes and 46 arcs This particular network described potential anesthesia problems in a hospital operating room In addition to the data, the program was also given an initial ordering over the 37 variables that was consistent with the partial

ordering of variable dependencies in the actual network The program succeeded in reconstructing the correct Bayesian network structure almost exactly, with the exception of one incorrectly deleted arc and one incorrectly added arc Constraint-based approaches to learning Bayesian network structure have also been developed (eg, Spirtes et al 1993) These approaches infer independence and dependence relationships from the data, and then use these relationships to construct Bayesian networks Surveys of current approaches to learning Bayesian networks are provided by Heckerman (1995) and Buntine (1994)