Machine Learning - DISCo

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4. The probability that the random variable R will take on a specific value r (e.g., the probability of observing exactly r heads) is given by the Binomial distribution n! Pr(R = r) = pr(l - p)"-' r!(n - r)! A plot of this probability distribution is shown in Table 5.3. The Binomial distribution characterizes the probability of observing r heads from n coin flip experiments, as well as the probability of observing r errors in a data sample containing n randomly drawn instances. 5.3.3 Mean and Variance Two properties of a random variable that are often of interest are its expected value (also called its mean value) and its variance. The expected value is_the average of the values taken on by repeatedly sampling the random variable. More precisely Definition: Consider a random variable Y that takes on the possible values yl, . . . yn. The expected value of Y, E[Y], is For example, if Y takes on the value 1 with probability .7 and the value 2 with probability .3, then its expected value is (1 .0.7 + 2.0.3 = 1.3). In case the random variable Y is governed by a Binomial distribution, then it can be shown that E [Y] = np (5.4) where n and p are the parameters of the Binomial distribution defined in Equation (5.2). A second property, the variance, captures the "width or "spread" of the probability distribution; that is, it captures how far the random variable is expected to vary from its mean value. Definition: The variance of a random variable Y, Var[Y], is Var[Y] = E[(Y - E[Y])~] (5.5) The variance describes the expected squared error in using a single observation of Y to estimate its mean E[Y]. The square root of the variance is called the standard deviation of Y, denoted oy . Definition: The standard deviation of a random variable Y, uy, is
In case the random variable Y is governed by a Binomial distribution, then the variance and standard deviation are given by 5.3.4 Estimators, Bias, and Variance Now that we have shown that the random variable errors(h) obeys a Binomial distribution, we return to our primary question: What is the likely difference between errors(h) and the true error errorv(h)? Let us describe errors(h) and errorv(h) using the terms in Equation (5.2) defining the Binomial distribution. We then have where n is the number of instances in the sample S, r is the number of instances from S misclassified by h, and p is the probability of misclassifying a single instance drawn from 23. Statisticians call errors(h) an estimator for the true error errorv(h). In general, an estimator is any random variable used to estimate some parameter of the underlying population from which the sample is drawn. An obvious question to ask about any estimator is whether on average it gives the right estimate. We define the estimation bias to be the difference between the expected value of the estimator and the true value of the parameter. Definition: The estimation bias of an estimator Y for an arbitrary parameter p is If the estimation bias is zero, we say that Y is an unbiased estimator for p. Notice this will be the case if the average of many random values of Y generated by repeated random experiments (i.e., E[Y]) converges toward p. Is errors(h) an unbiased estimator for errorv(h)? Yes, because for a Binomial distribution the expected value of r is equal to np (Equation r5.41). It follows, given that n is a constant, that the expected value of rln is p. Two quick remarks are in order regarding the estimation bias. First, when we mentioned at the beginning of this chapter that testing the hypothesis on the training examples provides an optimistically biased estimate of hypothesis error, it is exactly this notion of estimation bias to which we were referring. In order for errors(h) to give an unbiased estimate of errorv(h), the hypothesis h and sample S must be chosen independently. Second, this notion of estimation bias should not be confused with the inductive bias of a learner introduced in Chapter 2. The
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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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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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