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R~~~~ss(Frontier, Rule, Literal, &i) Frontier: Set of literals to be regressed through Rule Rule: A Horn clause Literal: A literal in Frontier that is inferred by Rule in the explanation Oki: The substitution that unijies the head of Rule to the corresponding literal in the explanation Returns the set of literals forming the weakest preimage of Frontier with respect to Rule head t head of Rule body t body of Rule Bkl t the most general unifier of head with Literal such that there exists a substitution Bli for which Ori (Bkl (head)) = Bhi (head) Return Okl (Frontier - head + body) Example (the bottommost regression step in Figure 11.3): h?~~~ss(Frontier, Rule, Literd, @hi) where Frontier = {Volume(x, us), Density(x, dx), Equal(wx, times(vx,dx)), LessThan(wx, wy), Weight(y, wy)) Rule = Weight(z, 5) c Type(z, Endtable) Literal = Weight(y, wy) 6ki = {z/Obj21 head c Weight (z, 5) body c Type(z, Endtable) Bhl e {z/y, wy/5], where Bri = (ylObj2) Return {Volume(x, us), Density(x, dx), Equal (wx, times(vx, dx)), LessThan(wx, 5). Type(y, Endtable)] TABLE 11.3 Algorithm for regressing a set of literals through a single Horn clause. The set of literals given by Frontier is regressed through Rule. Literal is the member of Frontier inferred by Rule in the explanation. The substitution Bki gives the binding of variables from the head of Rule to the corresponding literal in the explanation. The algorithm first computes a substitution Bhl that unifies the Rule head to Literal, in a way that is consistent with the substitution Bki. It then applies this substitution Oh[ to construct the preimage of Frontier with respect to Rule. The symbols "+" and "-" in the algorithm denote set union and set difference. The notation {zly] denotes the substitution of y in place of z. An example trace is given. explanation applies to only a special case of the target concept. As noted earlier, for the current example the final rule is SafeToStack(x, y) t Volume(x, vx) A Density(x, d x ) ~ Equal(wx, times(vx, dx)) A LessThan(wx, 5 ) ~ Type(y, Endtable) 11.2.1.3 REFINE THE CURRENT HYPOTHESIS The current hypothesis at each stage consists of the set of Horn clauses learned thus far. At each stage, the sequential covering algorithm picks a new positive
example that is not yet covered by the current Horn clauses, explains this new example, and formulates a new rule'according to the procedure described above. Notice only positive examples are covered in the algorithm as we have defined it, and the learned set of Horn clause rules predicts only positive examples. A new instance is classified as negative if the current rules fail to predict that it is positive. This is in keeping with the standard negation-as-failure approach used in Horn clause inference systems such as PROLOG. 11.3 REMARKS ON EXPLANATION-BASED LEARNING As we saw in the above example, PROLOG-EBG conducts a detailed analysis of individual training examples to determine how best to generalize from the specific example to a general Horn clause hypothesis. The following are the key properties of this algorithm. 0 Unlike inductive methods, PROLOG-EBG produces justified general hypotheses by using prior knowledge to analyze individual examples. 0 The explanation of how the example satisfies the target concept determines which example attributes are relevant: those mentioned by the explanation. 0 The further analysis of the explanation, regressing the target concept to determine its weakest preimage with respect to the explanation, allows deriving more general constraints on the values of the relevant features. 0 Each learned Horn clause corresponds to a sufficient condition for satisfying the target concept. The set of learned Horn clauses covers the positive training examples encountered by the learner, as well as other instances that share the same explanations. 0 The generality of the learned Horn clauses will depend on the formulation of the domain theory and on the sequence in which training examples are considered. 0 PROLOG-EBG implicitly assumes that the domain theory is correct and complete. If the domain theory is incorrect or incomplete, the resulting learned concept may also be incorrect. There are several related perspectives on explanation-based learning that help to understand its capabilities and limitations. 0 EBL as theory-guided generalization of examples. EBL uses its given domain theory to generalize rationally from examples, distinguishing the relevant example attributes from the irrelevant, thereby allowing it to avoid the bounds on sample complexity that apply to purely inductive learning. This is the perspective implicit in the above description of the PROLOG-EBG algorithm. 0 EBL as example-guided reformulation of theories. The PROLOG-EBG algorithm can be viewed as a method for reformulating the domain theory into a more operational form. In particular, the original domain theory is reformulated by creating rules that (a) follow deductively from the domain theory,
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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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CHAFER 2 CONCEm LEARNING AND THE GE
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When learning the target concept, t
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CHAPTER 2 CONCEPT LEARNING AND THE
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is needed. In the general case, as
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2.5 VERSION SPACES AND THE CANDIDAT
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{ 1 FIGURE 2.3 A version space w
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CHAPTER 2 CONCEET LEARNJNG AND THE
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C H m R 2 CONCEPT LEARNING AND THE
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CH.4PTF.R 2 CONCEFT LEARNING AND TH
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the power set of X? In general, the
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CHAFI%R 2 CONCEPT LEARNING AND THE
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CHAPTER 2 CONCEPT. LEARNING AND THE
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CHAFTER 2 CONCEPT LEARNING AND THE
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Hirsh, H. (1991). Theoretical under
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CHAPTER 3 DECISION TREE LEARNING 53
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CHAPTER 3 DECISION TREE LEARMNG 55
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High wx CHAPTER 3 DECISION TREE LEA
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{Dl, D2, ..., Dl41 P+S-I Which attr
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3.6 INDUCTIVE BIAS IN DECISION TREE
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3.6.2 Why Prefer Short Hypotheses?
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easonable strategy, in fact it can
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Although the first of these approac
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multiple ways, then averaging the r
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CHAFER 3 DECISION TREE LEARNING 77
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C m 3 DECISION TREE LEARNING 79 Fay
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CHAPTER ARTIFICIAL NEURAL NETWORKS
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at normal speeds on public highways
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~t is also applicable to problems f
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FIGURE 43 A perceptron. A single pe
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this example. Notice the training r
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Gradient descent search determines
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- - CHAF'l'ER 4 ARTIFICIAL NEURAL N
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C H m R 4 ARTIFICIAL NEURAL NETWORK
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CHAPTER 4 ARTIFICIAL NEURAL NETWORK
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and combining this with Equations (
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Inputs Outputs Input 10000000 0 100
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FIGURE 4.8 Learning the 8 x 3 x 8 N
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30 x 32 resolution input images lef
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left, (0,1,0,O) to encode a face lo
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close to zero, as the bright face w
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4.8.2 Alternative Error Minimizatio
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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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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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Machine Learning - DISCo

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