Machine Learning - DISCo

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FIGURE 6.4 Instances generated by a mixture of two Normal distributions with identical variance a. The 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. 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 of a single Normal distribution given the observed data instances XI, x2, ..., 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 6.4, Equation (6.6), 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 (6.6) using our current notation, we have In this case, the sum of squared errors is minimized by the sample mean Our problem here, however, involves a mixture of k different Normal distributions, and we cannot observe which instances were generated by which distribution. Thus, we have a prototypical example of a problem involving hidden variables. In the example of Figure 6.4, we can think of the full description of each instance as the triple (xi, zil , ziz), where xi is the observed value of the ith instance and where zil and zi2 indicate which of the two Normal distributions was used to generate the value xi. In particular, zij has the value 1 if xi was created by the jth Normal distribution and 0 otherwise. Here xi is the observed variable in the description of the instance, and zil and zi2 are hidden variables. If the values of zil and zi2 were observed, we could use Equation (6.27) to solve for the means p1 and p2. Because they are not, we will instead use the EM algorithm. Applied to our k-means problem the EM algorithm searches for a maximum likelihood hypothesis by repeatedly re-estimating the expected values of the hidden variables zij given its current hypothesis (pI. .. pk), then recalculating the
CHAPTER 6 BAYESIAN LEARNING 193 maximum likelihood hypothesis using these expected values for the hidden variables. We will first describe this instance of the EM algorithm, and later state the EM algorithm in its general form. ' Applied to the problem of estimating the two means for Figure 6.4, the EM algorithm first initializes the hypothesis to h = (PI, p2), where p1 and p2 are arbitrary initial values. It then iteratively re-estimates h by repeating the following two steps until the procedure converges to a stationary value for h. Step 1: Calculate the expected value E[zij] of each hidden variable zi,, assuming the current hypothesis h = (p1, p2) holds. Step 2: Calculate a new maximum likelihood hypothesis h' = (pi, p;), assuming the value taken on by each hidden variable zij is its expected value E[zij] calculated in Step 1. Then replace the hypothesis h = (pl, p2) by the new hypothesis h' = (pi, pi) and iterate. Let us examine how both of these steps can be implemented in practice. / Step 1 must calculate the expected value of each zi,. This E[4] is just the probability that instance xi was generated by the jth Normal distribution Thus the first step is implemented by substituting the current values (pl, p2) and the observed xi into the above expression. In the second step we use the E[zij] calculated during Step 1 to derive a new maximum likelihood hypothesis h' = (pi, pi). AS we will discuss later, the maximum likelihood hypothesis in this case is given by Note this expression is similar to the sample mean from Equation (6.28) that is used to estimate p for a single Normal distribution. Our new expression is just the weighted sample mean for pj, with each instance weighted by the expectation E[z,j] that it was generated by the jth Normal distribution. The above algorithm for estimating the means of a mixture of k Normal distributions illustrates the essence of the EM approach: The current hypothesis is used to estimate the unobserved variables, and the expected values of these variables are then used to calculate an improved hypothesis. It can be proved that on each iteration through this loop, the EM algorithm increases the likelihood P(Dlh) unless it is at a local maximum. The algorithm thus converges to a local maximum likelihood hypothesis for (pl, w2).
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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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Page 171 and 172: . - Product rule: probability P(A A
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Page 217 and 218: C Instance space X Where c and h di
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Page 235 and 236: We define the optimal mistake bound
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CHAPTER 8 INSTANCE-BASED LEARMNG 24
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CHAPTER 8 INSTANCE-BASED LEARNING 2
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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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