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CHAPTER 6 BAYESIAN LEARNING
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In this section we introduce the key concepts and the representation of Bayesian belief networks More detailed treatments are given by Pearl (1988), Russell and Norvig (1995), Heckerman et al (1995), and Jensen (1996) In general, a Bayesian belief network describes the probability distribution over a set of variables Consider an arbitrary set of random variables Yl Y,, where each variable Yi can take on the set of possible values V(Yi) We define the joint space of the set of variables Y to be the cross product V(Yl) x V(Y2) x V(Y,) In other words, each item in the joint space corresponds to one of the possible assignments of values to the tuple of variables (Yl Y,) The probability distribution over this joint' space is called the joint probability distribution The joint probability distribution specifies the probability for each of the possible variable bindings for the tuple (Yl Y,) A Bayesian belief network describes the joint probability distribution for a set of variables
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6111 Conditional Independence
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Let us begin our discussion of Bayesian belief networks by defining precisely the notion of conditional independence Let X , Y, and Z be three discrete-valued random variables We say that X is conditionally independent of Y given Z if the probability distribution governing X is independent of the value of Y given a value for 2; that is, if
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where xi E V(X), yj E V(Y), and z k E V(Z) We commonly write the above expression in abbreviated form as P(XIY, Z ) = P(X1Z) This definition of conditional independence can be extended to sets of variables as well We say that the set of variables X1 Xi is conditionally independent of the set of variables Yl Ym given the set of variables 2 1 Z, if
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P ( X 1 XIJY1Ym, z Z,) = P ( X l X1]Z1 Z,) 1
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Note the correspondence between this definition and our use of conditional independence in the definition of the naive Bayes classifier The naive Bayes classifier assumes that the instance attribute A1 is conditionally independent of instance attribute A2 given the target value V This allows the naive Bayes classifier to calculate P ( A l , A21V) in Equation (620) as follows
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Equation (623) is just the general form of the product rule of probability from Table 61 Equation (624) follows because if A1 is conditionally independent of A2 given V, then by our definition of conditional independence P (A1 IA2, V ) = P(A1IV)
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S,B S,-B -C
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Campfire
FIGURE 63 A Bayesian belief network The network on the left represents a set of conditional independence assumptions In particular, each node is asserted to be conditionally independent of its nondescendants, given its immediate parents Associated with each node is a conditional probability table, which specifies the conditional distribution for the variable given its immediate parents in the graph The conditional probability table for the Campjire node is shown at the right, where Campjire is abbreviated to C , Storm abbreviated to S, and BusTourGroup abbreviated to B
6112 Representation
A Bayesian belief network (Bayesian network for short) represents the joint probability distribution for a set of variables For example, the Bayesian network in Figure 63 represents the joint probability distribution over the boolean variables Storm, Lightning, Thunder, ForestFire, Campjre, and BusTourGroup In general, a Bayesian network represents the joint probability distribution by specifying a set of conditional independence assumptions (represented by a directed acyclic graph), together with sets of local conditional probabilities Each variable in the joint space is represented by a node in the Bayesian network For each variable two types of information are specified First, the network arcs represent the assertion that the variable is conditionally independent of its nondescendants in the network given its immediate predecessors in the network We say Xjis a descendant of , Y if there is a directed path from Y to X Second, a conditional probability table is given for each variable, describing the probability distribution for that variable given the values of its immediate predecessors The joint probability for any desired assignment of values ( y l , , y,) to the tuple of network variables (YI Y,) can be computed by the formula
~ ( Y I , , yd = n p ( y i ~ p a r e n t s ( ~ i ) )
where Parents(Yi) denotes the set of immediate predecessors of Yi in the netParents(Yi)) are precisely the values stored in the work Note the values of P(yiJ conditional probability table associated with node Yi To illustrate, the Bayesian network in Figure 63 represents the joint probability distribution over the boolean variables Storm, Lightning, Thunder, Forest-