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【PDF】《Machine learning A Probabilistic Perspective》 MLAPP;by Kevin Murphy

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完整版,带目录,机器学习必备经典;大部头要用力啃。Machine learning A Probabilistic PerspectiveMachine LearningA Probabilistic PerspectiveKevin P. MurphyThe mit PressCambridge, MassachusettsLondon, Englando 2012 Massachusetts Institute of TechnologyAll rights reserved. No part of this book may be reproduced in any form by any electronic or mechanicalmeans(including photocopying, recording, or information storage and retrieval)without permission inwriting from the publisherFor information about special quantity discounts, please email special_sales@mitpress. mit. eduThis book was set in the HEx programming language by the author. Printed and bound in the UnitedStates of AmLibrary of Congress Cataloging-in-Publication InformationMurphy, Kevin Png:a piobabilistctive/Kevin P. Murphyp. cm. -(Adaptive computation and machine learning series)Includes bibliographical references and indexisBn 978-0-262-01802-9 (hardcover: alk. paper1. Machine learning. 2. Probabilities. I. TitleQ325.5M872012006.31-dc232012004558109876This book is dedicated to alessandro, Michael and stefanoand to the memory of gerard Joseph murphyContentsPreactXXVII1 IntroductionMachine learning: what and why?1..1Types of machine learning1.2 Supervised learning1.2.1Classification 31.2.2 Regression 83 Unsupervised learning 91.3.11.3.2Discovering latent factors 111.3.3 Discovering graph structure 131.3.4 Matrix completion 141.4 Some basic concepts in machine learning 161.4.1Parametric vs non-parametric models 161.4.2 A simple non-parametric classifier: K-nearest neighbors 161.4.3 The curse of dimensionality 181.4.4 Parametric models for classification and regression 191.4.5Linear regression 191.4.6Logistic regression1.4.7 Overfitting 221.4.8Model selection1.4.9No free lunch theorem242 Probability2.1 Introduction 272.2 A brief review of probability theory 282. 2. 1 Discrete random variables 282. 2.2 Fundamental rules 282.2.3B292. 2. 4 Independence and conditional independence 302. 2. 5 Continuous random variable32CONTENTS2.2.6 Quantiles 332.2.7 Mean and variance 332.3 Some common discrete distributions 342.3.1The binomial and bernoulli distributions 342.3.2 The multinomial and multinoulli distributions 352. 3.3 The Poisson distribution 372.3.4 The empirical distribution 372.4 Some common continuous distributions 382.4.1 Gaussian (normal) distribution 382.4.2Dte pdf 392.4.3 The Laplace distribution 412.4.4 The gamma distribution 412.4.5 The beta distribution 422.4.6 Pareto distribution2.5 Joint probability distributions 442.5.1Covariance and correlation442.5.2 The multivariate gaussian2.5.3 Multivariate Student t distribution 462.5.4 Dirichlet distribution 472.6 Transformations of random variables 492. 6. 1 Linear transformations 492.6.2 General transformations 502.6.3 Central limit theorem 512.7 Monte Carlo approximation 522.7.1 Example: change of variables, the MC way 532.7.2 Example: estimating T by Monte Carlo integration2.7.3 Accuracy of Monte Carlo approximation 542.8 Information theory562.8.1Entropy2.8.2 KL dive572.8.3 Mutual information 593 Generative models for discrete data 653.1 Introducti653.2 Bayesian concept learning 653.2.1Likelihood673.2.2 Prior 673.2.3P683.2.4Postedictive distribution3.2.5 A more complex prior 723.3 The beta-binomial model 723.3.1 Likelihood 733.3.2Prior743.3.3 Poster3.3.4Posterior predictive distributionCONTENTS3.4 The Dirichlet-multinomial model 783. 4. 1 Likelihood 793.4.2 Prior 793.4.3 Posterior 793.4.4Posterior predictive813.5 Naive Bayes classifiers 823.5.1 Model fitting 833.5.2 Using the model for prediction 853.5.3 The log-sum-exp trick 803.5.4 Feature selection using mutual information 863.5.5 Classifying documents using bag of words 84 Gaussian models4.1 Introduction974.1.1Notation974. 1.2 Basics 974. 1.3 MlE for an mvn 994.1.4 Maximum entropy derivation of the gaussian 1014.2 Gaussian discriminant analysis 1014.2.1 Quadratic discriminant analysis(QDA) 1024.2.2 Linear discriminant analysis (LDA) 1034.2.3 Two-claSs LDA 1044.2.4 MLE for discriminant analysis 1064.2.5 Strategies for preventing overfitting 1064.2.6 Regularized LDA* 104.2.7 Diagonal LDA4.2.8 Nearest shrunken centroids classifier1094.3 Inference in jointly Gaussian distributions 1104.3.1Statement of the result 1114.3.2 Examples4.3.3 Information form 1154.3.4 Proof of the result 1164.4 Linear Gaussian systems 1194.4.1Statement of the result 1194.4.2 Examples 1204.4.3 Proof of the result1244.5 Digression: The Wishart distribution4.5. 1 Inverse Wishart distribution 1264.5.2 Visualizing the wishart distribution* 1274.6 Inferring the parameters of an MVn 1274.6.1 Posterior distribution of u 1284.6.2 Posterior distribution of e1284.6.3 Posterior distribution of u and 2* 1324.6.4 Sensor fusion with unknown precisions 138

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    在学习代数学之余,值得一看的代数学书籍。里面介绍了更为丰富的代数学概念和结论。PREFACEMy purpose in this book is to treat linear transformations on finite-dimensional vector spaces by the methods of more general theories. Theidea is to emphasize the simple geometric notions common to many partsof mathematics and its applications, and to do so in a language that givesaway the trade secrets and tells the student what is in the back of the mindsof people proving theorems about integral equations and Hilbert spaces.The reader does not, however, have to share my prejudiced motivationExcept for an occasional reference to undergraduate mathematics the bookis self-contained and may be read by anyone who is trying to get a feelingfor the linear problems usually discussed in courses on matrix theory orhigher"algebra. The algebraic, coordinate-free methods do not lose powerand elegance by specialization to a finite number of dimensions, and theyare, in my belief, as elementary as the classical coordinatized treatmentI originally intended this book to contain a theorem if and only if aninfinite-dimensional generalization of it already exists, The temptingeasiness of some essentially finite-dimensional notions and results washowever, irresistible, and in the final result my initial intentions are justbarely visible. They are most clearly seen in the emphasis, throughout, ongeneralizable methods instead of sharpest possible results. The reader maysometimes see some obvious way of shortening the proofs i give In suchcases the chances are that the infinite-dimensional analogue of the shorterproof is either much longer or else non-existent.A preliminary edition of the book (Annals of Mathematics Studies,Number 7, first published by the Princeton University Press in 1942)hasbeen circulating for several years. In addition to some minor changes instyle and in order, the difference between the preceding version and thisone is that the latter contains the following new material:(1) a brief dis-cussion of fields, and, in the treatment of vector spaces with inner productsspecial attention to the real case.(2)a definition of determinants ininvariant terms, via the theory of multilinear forms. 3 ExercisesThe exercises(well over three hundred of them) constitute the mostsignificant addition; I hope that they will be found useful by both studentPREFACEand teacher. There are two things about them the reader should knowFirst, if an exercise is neither imperative "prove that.., )nor interrogtive("is it true that...?" )but merely declarative, then it is intendedas a challenge. For such exercises the reader is asked to discover if theassertion is true or false, prove it if true and construct a counterexample iffalse, and, most important of all, discuss such alterations of hypothesis andconclusion as will make the true ones false and the false ones true. Secondthe exercises, whatever their grammatical form, are not always placed 8oas to make their very position a hint to their solution. Frequently exer-cises are stated as soon as the statement makes sense, quite a bit beforemachinery for a quick solution has been developed. A reader who tries(even unsuccessfully) to solve such a"misplaced"exercise is likely to ap-preciate and to understand the subsequent developments much better forhis attempt. Having in mind possible future editions of the book, I askthe reader to let me know about errors in the exercises, and to suggest im-provements and additions. (Needless to say, the same goes for the text.)None of the theorems and only very few of the exercises are my discovery;most of them are known to most working mathematicians, and have beenknown for a long time. Although i do not give a detailed list of my sources,I am nevertheless deeply aware of my indebtedness to the books and papersfrom which I learned and to the friends and strangers who, before andafter the publication of the first version, gave me much valuable encourage-ment and criticism. Iam particularly grateful to three men: J. L. Dooband arlen Brown, who read the entire manuscript of the first and thesecond version, respectively, and made many useful suggestions, andJohn von Neumann, who was one of the originators of the modern spiritand methods that I have tried to present and whose teaching was theinspiration for this bookP、R.HCONTENTS的 FAPTERPAGRI SPACESI. Fields, 1; 2. Vector spaces, 3; 3. Examples, 4;4. Comments, 55. Linear dependence, 7; 6. Linear combinations. 9: 7. Bases, 108. Dimension, 13; 9. Isomorphism, 14; 10. Subspaces, 16; 11. Calculus of subspaces, 17; 12. Dimension of a subspace, 18; 13. Dualspaces, 20; 14. Brackets, 21; 15. Dual bases, 23; 16. Reflexivity, 24;17. Annihilators, 26; 18. Direct sums, 28: 19. Dimension of a directsum, 30; 20. Dual of a direct sum, 31; 21. Qguotient spaces, 33;22. Dimension of a quotient space, 34; 23. Bilinear forms, 3524. Tensor products, 38; 25. Product bases, 40 26. Permutations41; 27. Cycles,44; 28. Parity, 46; 29. Multilinear forms, 4830. Alternating formB, 50; 31. Alternating forms of maximal degree,52II. TRANSFORMATIONS32. Linear transformations, 55; 33. Transformations as vectors, 5634. Products, 58; 35. Polynomials, 59 36. Inverses, 61; 37. Mat-rices, 64; 38. Matrices of transformations, 67; 39. Invariance,7l;40. Reducibility, 72 41. Projections, 73 42. Combinations of pro-jections, 74; 43. Projections and invariance, 76; 44. Adjoints, 78;45. Adjoints of projections, 80; 46. Change of basis, 82 47. Similarity, 84; 48. Quotient transformations, 87; 49. Range and null-space, 88; 50. Rank and nullity, 90; 51. Transformations of rankone, 92 52. Tensor products of transformations, 95; 53. Determinants, 98 54. Proper values, 102; 55. Multiplicity, 104; 56. Triangular form, 106; 57. Nilpotence, 109; 58. Jordan form. 112III ORTHOGONALITY11859. Inner products, 118; 60. Complex inner products, 120; 61. Innerproduct spaces, 121; 62 Orthogonality, 122; 63. Completeness, 124;64. Schwarz e inequality, 125; 65. Complete orthonormal sets, 127;CONTENTS66. Projection theorem, 129; 67. Linear functionals, 130; 68. P aren, gBCHAPTERtheses versus brackets, 13169. Natural isomorphisms, 138;70. Self-adjoint transformations, 135: 71. Polarization, 13872. Positive transformations, 139; 73. Isometries, 142; 74. Changeof orthonormal basis, 144; 75. Perpendicular projections, 14676. Combinations of perpendicular projections, 148; 77. Com-plexification, 150; 78. Characterization of spectra, 158; 79. Spec-ptral theorem, 155; 80. normal transformations, 159; 81. Orthogonaltransformations, 162; 82. Functions of transformations, 16583. Polar decomposition, 169; 84. Commutativity, 171; 85. Self-adjoint transformations of rank one, 172IV. ANALYSIS....17586. Convergence of vectors, 175; 87. Norm, 176; 88. Expressions forthe norm, 178; 89. bounds of a self-adjoint transformation, 17990. Minimax principle, 181; 91. Convergence of linear transformations, 182 92. Ergodic theorem, 184 98. Power series, 186APPENDIX. HILBERT SPACERECOMMENDED READING, 195INDEX OF TERMS, 197INDEX OF SYMBOLS, 200CHAPTER ISPACES§L. FieldsIn what follows we shall have occasion to use various classes of numbers(such as the class of all real numbers or the class of all complex numbers)Because we should not at this early stage commit ourselves to any specificclass, we shall adopt the dodge of referring to numbers as scalars. Thereader will not lose anything essential if he consistently interprets scalarsas real numbers or as complex numbers in the examples that we shallstudy both classes will occur. To be specific(and also in order to operateat the proper level of generality) we proceed to list all the general factsabout scalars that we shall need to assume(A)To every pair, a and B, of scalars there corresponds a scalar a+called the sum of a and B, in such a way that(1) addition is commutative,a+β=β+a,(2)addition is associative, a+(8+y)=(a+B)+y(3 there exists a unique scalar o(called zero)such that a+0= a forevery scalar a, and(4)to every scalar a there corresponds a unique scalar -a such that十(0(B)To every pair, a and B, of scalars there corresponds a scalar aBcalled the product of a and B, in such a way that(1)multiplication is commutative, aB pa(2)multiplication is associative, a(Br)=(aB)Y,( )there exists a unique non-zero scalar 1 (called one)such that al afor every scalar a, and(4)to every non-zero scalar a there corresponds a unique scalar a-1or-such that aaSPACES(C)Multiplication is distributive with respect to addition, a(a+n)If addition and multiplication are defined within some set of objectsscalars) so that the conditions(A),B), and (c)are satisfied, then thatset(together with the given operations) is called a field. Thus, for examplethe set Q of all rational numbers(with the ordinary definitions of sumand product)is a field, and the same is true of the set of all real numberaand the set e of all complex numbersHHXERCISIS1. Almost all the laws of elementary arithmetic are consequences of the axiomsdefining a field. Prove, in particular, that if 5 is field and if a, and y belongto 5. then the following relations hold80+a=ab )Ifa+B=a+r, then p=yca+(B-a)=B (Here B-a=B+(a)(d)a0=0 c=0.(For clarity or emphasis we sometimes use the dot to indi-cate multiplication.()(-a)(-p)(g).If aB=0, then either a=0 or B=0(or both).2.(a)Is the set of all positive integers a field? (In familiar systems, such as theintegers, we shall almost always use the ordinary operations of addition and multi-lication. On the rare occasions when we depart from this convention, we shallgive ample warningAs for "positive, "by that word we mean, here and elsewherein this book, "greater than or equal to zero If 0 is to be excluded, we shall say"strictly positive(b)What about the set of all integers?(c) Can the answers to these questiong be changed by re-defining addition ormultiplication (or both)?3. Let m be an integer, m2 2, and let Zm be the set of all positive integers lessthan m, zm=10, 1, .. m-1). If a and B are in Zmy let a +p be the leastpositive remainder obtained by dividing the(ordinary) sum of a and B by m, andproduct of a and B by m.(Example: if m= 12, then 3+11=2 and 3. 11=9)a) Prove that i is a field if and only if m is a prime.(b What is -1 in Z5?(c) What is囊izr?4. The example of Z, (where p is a prime)shows that not quite all the laws ofelementary arithmetic hold in fields; in Z2, for instance, 1 +1 =0. Prove thatif is a field, then either the result of repeatedly adding 1 to itself is always dif-ferent from 0, or else the first time that it is equal to0 occurs when the numberof summands is a prime. (The characteristic of the field s is defined to be 0 in thefirst case and the crucial prime in the second)SEC. 2VECTOR SPACES35. Let Q(v2)be the set of all real numbers of the form a+Bv2, wherea and B are rational.(a)Ie(√2) a field?(b )What if a and B are required to be integer?6.(a)Does the set of all polynomials with integer coefficients form a feld?(b)What if the coeficients are allowed to be real numbers?7: Let g be the set of all(ordered) pairs(a, b)of real numbers(a) If addition and multiplication are defined by(a月)+(,6)=(a+y,B+6)and(a,B)(Y,8)=(ary,B6),does s become a field?(b )If addition and multiplication are defined by(α,月)+⑦,b)=(a+%,B+6)daB)(,b)=(ay-6a6+的y),is g a field then?(c)What happens (in both the preceding cases)if we consider ordered pairs ofcomplex numbers instead?§2. Vector spaceWe come now to the basic concept of this book. For the definitionthat follows we assume that we are given a particular field s; the scalarsto be used are to be elements of gDEFINITION. 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Therelation between a vector space V and the underlying field s is usuallydescribed by saying that v is a vector space over 5. If S is the field Rof real number, u is called a real vector space; similarly if s is Q or if gise, we speak of rational vector spaces or complex vector space§3. ExamplesBefore discussing the implications of the axioms, we give some examplesWe shall refer to these examples over and over again, and we shall use thenotation established here throughout the rest of our work.(1) Let e(= e)be the set of all complex numbers; if we interpretr+y and az as ordinary complex numerical addition and multiplicatione becomes a complex vector space2)Let o be the set of all polynomials, with complex coeficients, in avariable t. 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