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CHAPTER 6 BAYESIAN LEARNING
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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)
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612 THE EM ALGORITHM
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In many practical learning settings, only a subset of the relevant instance features might be observable For example, in training or using the Bayesian belief network of Figure 63, we might have data where only a subset of the network variables Storm, Lightning, Thunder, ForestFire, Campfire, and BusTourGroup have been observed Many approaches have been proposed to handle the problem of learning in the presence of unobserved variables As we saw in 3, if some variable is sometimes observed and sometimes not, then we can use the cases for which it has been observed to learn to predict its values when it is not In this section we describe the EM algorithm (Dempster et al 1977), a widely used approach to learning in the presence of unobserved variables The EM algorithm can be used even for variables whose value is never directly observed, provided the general form of the probability distribution governing these variables is known The EM algorithm has been used to train Bayesian belief networks (see Heckerman 1995) as well as radial basis function networks discussed in Section 84 The EM algorithm is also the basis for many unsupervised clustering algorithms (eg, Cheeseman et al 1988), and it is the basis for the widely used Baum-Welch forward-backward algorithm for learning Partially Observable Markov Models (Rabiner 1989)
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6121 Estimating Means of k Gaussians
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The easiest way to introduce the EM algorithm is via an example Consider a problem in which the data D is a set of instances generated by a probability distribution that is a mixture of k distinct Normal distributions This problem setting is illustrated in Figure 64 for the case where k = 2 and where the instances are the points shown along the x axis Each instance is generated using a two-step process First, one of the k Normal distributions is selected at random Second, a single random instance xi is generated according to this selected distribution This process is repeated to generate a set of data points as shown in the figure To simplify our discussion, we consider the special case where the selection of the single Normal distribution at each step is based on choosing each with uniform probability, where each of the k Normal distributions has the same variance a2,and where a2 is known The learning task is to output a hypothesis h = (FI, pk) that describes the means of each of the k distributions We would like to find
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FIGURE 64 Instances generated by a mixture of two Normal distributions with identical variance aThe instances are shown by the points along the x axis If the means of the Normal distributions are unknown, the EM algorithm can be used to search for their maximum likelihood estimates
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a maximum likelihood hypothesis for these means; that is, a hypothesis h that maximizes p ( D lh) Note it is easy to calculate the maximum likelihood hypothesis for the mean x2, of a single Normal distribution given the observed data instances X I , , xm drawn from this single distribution This problem of finding the mean of a single distribution is just a special case of the problem discussed in Section 64, Equation (66), where we showed that the maximum likelihood hypothesis is the one that minimizes the sum of squared errors over the m training instances Restating Equation (66) using our current notation, we have
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