Machine Learning - DISCo

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set, minus 1.96 times the estimated standard deviation). The net effect is that for large data sets, the pessimistic estimate is very close to the observed accuracy (e.g., the standard deviation is very small), whereas it grows further from the observed accuracy as the size of the data set decreases. Although this heuristic method is not statistically valid, it has nevertheless been found useful in practice. See Chapter 5 for a discussion of statistically valid approaches to estimating means and confidence intervals. Why convert the decision tree to rules before pruning? There are three main advantages. Converting to rules allows distinguishing among the different contexts in which a decision node is used. Because each distinct path through the decision tree node produces a distinct rule, the pruning decision regarding that attribute test can be made differently for each path. In contrast, if the tree itself were pruned, the only two choices would be to remove the decision node completely, or to retain it in its original form. Converting to rules removes the distinction between attribute tests that occur near the root of the tree and those that occur near the leaves. Thus, we avoid messy bookkeeping issues such as how to reorganize the tree if the root node is pruned while retaining part of the subtree below this test. Converting to rules improves readability. Rules are often easier for to understand. 3.7.2 Incorporating Continuous-Valued Attributes Our initial definition of ID3 is restricted to attributes that take on a discrete set of values. First, the target attribute whose value is predicted by the learned tree must be discrete valued. Second, the attributes tested in the decision nodes of the tree must also be discrete valued. This second restriction can easily be removed so that continuous-valued decision attributes can be incorporated into the learned tree. This can be accomplished by dynamically defining new discretevalued attributes that partition the continuous attribute value into a discrete set of intervals. In particular, for an attribute A that is continuous-valued, the algorithm can dynamically create a new boolean attribute A, that is true if A < c and false otherwise. The only question is how to select the best value for the threshold c. As an example, suppose we wish to include the continuous-valued attribute Temperature in describing the training example days in the learning task of Table 3.2. Suppose further that the training examples associated with a particular node in the decision tree have the following values for Temperature and the target attribute PlayTennis. Temperature: 40 48 60 72 80 90 PlayTennis: No No Yes Yes Yes NO
CHAPTER 3 DECISION TREE LEARNING 73 What threshold-based boolean attribute should be defined based on Temperature? Clearly, we would like to pick a threshold, c, that produces the greatest information gain. By sorting the examples according to the continuous attribute A, then identifying adjacent examples that differ in their target classification, we can generate a set of candidate thresholds midway between the corresponding values of A. It can be shown that the value of c that maximizes information gain must always lie at such a boundary (Fayyad 1991). These candidate thresholds can then be evaluated by computing the information gain associated with each. In the current example, there are two candidate thresholds, corresponding to the values of Temperature at which the value of PlayTennis changes: (48 + 60)/2, and (80 + 90)/2. The information gain can then be computed for each of the candidate attributes, Temperat~re,~~ and Temperat~re,~~, and the best can be selected (Temperat~re,~~). This dynamically created boolean attribute can then compete with the other discrete-valued candidate attributes available for growing the decision tree. Fayyad and Irani (1993) discuss an extension to this approach that splits the continuous attribute into multiple intervals rather than just two intervals based on a single threshold. Utgoff and Brodley (1991) and Murthy et al. ( 1994) discuss approaches that define features by thresholding linear combinations of several continuous-valued attributes. 3.7.3 Alternative Measures for Selecting Attributes There is a natural bias in the information gain measure that favors attributes with many values over those with few values. As an extreme example, consider the attribute Date, which has a very large number of possible values (e.g., March 4, 1979). If we were to add this attribute to the data in Table 3.2, it would have the highest information gain of any of the attributes. This is because Date alone perfectly predicts the target attribute over the training data. Thus, it would be selected as the decision attribute for the root node of the tree and lead to a (quite broad) tree of depth one, which perfectly classifies the training data. Of course, this decision tree would fare poorly on subsequent examples, because it is not a useful predictor despite the fact that it perfectly separates the training data. What is wrong with the attribute Date? Simply put, it has so many possible values that it is bound to separate the training examples into very small subsets. Because of this, it will have a very high information gain relative to the training examples, despite being a very poor predictor of the target function over unseen instances. One way to avoid this difficulty is to select decision attributes based on some measure other than information gain. One alternative measure that has been used successfully is the gain ratio (Quinlan 1986). The gain ratio measure penalizes attributes such as Date by incorporating a term, called split informution, that is sensitive to how broadly and uniformly the attribute splits the data:
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Machine Learning Tom M. Mitchell Pr
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xvi PREFACE A third principle that
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CHAPTER INTRODUCTION Ever since com
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CHAPTER 1 INTRODUCITON 3 0 Learning
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CHAFTlB 1 INTRODUCTION 5 that allow
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1.2.2 Choosing the Target Function
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CHAPTER I INTRODUCTION 9 x5: the nu
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Thus, we seek the weights, or equiv
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could involve creating board positi
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the available training examples. Th
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0 Chapter 10 covers algorithms for
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ination of board features of your c
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Page 39 and 40: is needed. In the general case, as
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Page 63 and 64: Hirsh, H. (1991). Theoretical under
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Page 79 and 80: easonable strategy, in fact it can
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BACKPROPAGATION searches the space
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4.6. Explain informally why the del
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Mitchell, T. M., & Thrun, S. B. (19
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the impact of possible pruning step
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Here the notation Pr denotes that t
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a A random variable can be viewed a
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5.3.2 The Binomial Distribution A g
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In case the random variable Y is go
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which is wide enough to contain 95%
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intervals for discrete-valued hypot
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Then as n + co, the distribution go
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we might observe this difference in
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1. Partition the available data Do
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perfectly fits the form of the abov
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CHAFER 5 EVALUATING HYPOTHESES 151
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Geman, S., Bienenstock, E., & Dours
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CHAFER 6 BAYESIAN LEARNING 155 U Fo
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CHAPTER 6 BAYESIAN LEARNING 157 As
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. - Product rule: probability P(A A
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CHAPTER 6 BAYESIAN LEARNING 161 Rec
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CHAPTER 6 BAYESIAN LEARNING 163 of
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CHAPTER 6 BAYESIAN LEARNING 165 a l
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CHAPTER 6 BAYESIAN LEARNING 167 the
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CHAPTER 6 BAYESIAN LEARNING 169 doe
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CIUPlER 6 BAYESIAN LEARNING 171 sub
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CHAPTER 6 BAYESIAN LEARNING 173 0 T
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CHAFER 6 BAYESIAN LEARNING 175 the
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6.9 NAIVE BAYES CLASSIFIER CHAPTJZR
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CHAETER 6 BAYESIAN LEARNING 179 Sim
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-a- CHAPTER 6 BAYESIAN LEARNING 181
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CHAPTER 6 BAYESIAN LEARNING 183 Exa
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CHAPTER 6 BAYESIAN LEARNING 185 In
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C H m R 6 BAYESIAN LEARNING 187 Fir
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CHAPTER 6 BAYESIAN LEARNING 189 For
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CHAPTER 6 BAYESIAN LEARNING 191 ord
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CHAPTER 6 BAYESIAN LEARNING 193 max
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CHAPTER 6 BAYESIAN LEARNING 195 Ste
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6.13 SUMMARY AND FURTHER READING Th
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CHAPTER 6 BAYESIAN LEARNING 199 pro
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CHAPTER COMPUTATIONAL LEARNING THEO
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CHAPTER 7 COMPUTATIONAL LEARNING TH
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C Instance space X Where c and h di
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usually more concerned with the num
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are drawn. Haussler (1988) provides
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CHAPTER 7 COMPUTATIONAL LEARNING TH
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C that contains every teachable con
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Instance space X FIGURE 73 A set of
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can show that it is at least 3, as
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The following theorem bounds the VC
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FIND-S: C CHAPTER 7 COMPUTATIONAL L
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We define the optimal mistake bound
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log' Rearranging terms yields which
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Current research on computational l
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(b) You now draw a new set of 100 i
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CHAPTER 8 INSTANCE-BASED LEARNING 2
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(i.e., on all axes in the Euclidean
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8.3.1 Locally Weighted Linear Regre
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C H m R 8 INSTANCE-BASED LEARNING 2
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CHAPTER 8 INSTANCEBASED LEARNING 24
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CHAPTER 8 INSTANCE-BASED LEARMNG 24
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A thorough discussion of radial bas
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CHAPTER GENETIC ALGORITHMS Genetic
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Fitness: A function that assigns an
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can then be represented by the foll
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the crossover point n is chosen at
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value of the target attribute c. Th
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In the above experiment, the two ne
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The second step above follows from
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FIGURE 9.2 Crossover operation appl
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0 (DU x y) (do until), which execut
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9.6.2 Baldwin Effect Although Lamar
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iteration, the most fit members of
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Belew, R. K., & Mitchell, M. (Eds.)
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Tumey, P. D., Whitley, D., & Anders
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As an example of first-order rule s
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10.2.1 General to Specific Beam Sea
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A few remarks on the LEARN-ONE-RULE
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decisions regarding preconditions o
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10.4 LEARNING FIRST-ORDER RULES In
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Every well-formed expression is com
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eralize the current disjunctive hyp
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Here let us also make the closed wo
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In the case of noise-free training
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(e.g., neural network) fits the dat
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C : KnowMaterial v -Study C: KnowMa
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CHAPTER 10 LEARNING SETS OF RULES 2
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In contrast, the generate-and-test
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which enable the user to specify th
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Wrobel (1992) use rule schemata in
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De Raedt, L., & Bruynooghe, M. (199
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CHAPTER ANALYTICAL LEARNING Inducti
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What is interesting about this ches
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Given: rn Instance space X: Each in
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theories to automatically improve p
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Explanation: Training Example: FIGU
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FIGURE 11.3 Computing the weakest p
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example that is not yet covered by
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contain no description of such a pr
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additional detail that must be cons
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CHAPTER 11 ANALYTICAL LEARNING 325
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explaining to itself the reason for
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that fits the learner's prior knowl
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Height (Joe, Short) Height(Sue, Sho
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CHAF'TER 11 ANALYTICAL LEARNING 333
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CHAPTER 12 COMBINING INDUCTIVE AND
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CHAF'TER 12 COMBINING INDUCTIVE AND
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12.2.2 Hypothesis Space Search CAAP
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- KBANN(Domain-Theory, Training_Exa
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CHAF'TER 12 COMBINING INDUCTIVE AND
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Hypothesis Space Hypotheses that ma
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CHAFER 12 COMBINING INDUCTIVE AND A
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CHAPTER 12 COMBmG INDUCTIVE AND ANA
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Hypothesis Space 1 Hypotheses thatf
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y first order Horn clauses to learn
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Craven, M. W., & Shavlik, J. W. (19
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CHAPTER REINFORCEMENT LEARNING Rein
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has or does not have prior knowledg
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CHAPTER 13 REINFORCEMENT LEARNING 3
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CHAPTER 13 REINFORCEMENT LEARNJNG 3
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While Q learning and related reinfo
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a neural network reinforcement lear
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CHAFER 13 REINFORCEMENT LEARNING 38
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CHaPrER 13 REINFORCEMENT LEARNING 3
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APPENDIX NOTATION Below is a summar
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INDEXES
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in KBANN algorithm, 343-344 optimiz
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leave-one-out, 235 in neural networ
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extensions to, 258-259 ID5R algorit
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of LMS algorithm, 64 of ROTE-LEARNE
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prediction of probabilities with, 1
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Probability distribution, 133. See
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Split infomation, 73-74 Squashing f
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