Bayesian Reasoning and Machine Learning

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CONTENTS CONTENTS 3.2.2 Unreliable evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Belief Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3.1 Conditional independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3.2 The impact of collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.3 Graphical path manipulations for independence . . . . . . . . . . . . . . . . . . . . . . 43 3.3.4 d-Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.5 Graphical and distributional in/dependence . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3.6 Markov equivalence in belief networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3.7 Belief networks have limited expressibility . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 Simpson’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.2 The do-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.3 Influence diagrams and the do-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6.1 Naive inference demo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6.2 Conditional independence demo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6.3 Utility routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4 Graphical Models 57 4.1 Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Markov Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2.1 Markov properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2.2 Markov random fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2.3 Hammersley-Clifford Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.4 Conditional independence using Markov networks . . . . . . . . . . . . . . . . . . . . . 63 4.2.5 Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Chain Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.4 Factor Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.1 Conditional independence in factor graphs . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.5 Expressiveness of Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.7 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5 Efficient Inference in Trees 75 5.1 Marginal Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.1 Variable elimination in a Markov chain and message passing . . . . . . . . . . . . . . . 75 5.1.2 The sum-product algorithm on factor graphs . . . . . . . . . . . . . . . . . . . . . . . 78 5.1.3 Dealing with Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.1.4 Computing the marginal likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.1.5 The problem with loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Other Forms of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.1 Max-Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.2 Finding the N most probable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.3 Most probable path and shortest path . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2.4 Mixed inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Inference in Multiply Connected Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3.1 Bucket elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.3.2 Loop-cut conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.4 Message Passing for Continuous Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.6 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.6.1 Factor graph examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.6.2 Most probable and shortest path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 XII DRAFT January 9, 2013
CONTENTS CONTENTS 5.6.3 Bucket elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.6.4 Message passing on Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6 The Junction Tree Algorithm 97 6.1 Clustering Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.1.1 Reparameterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Clique Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.2.1 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2.2 Absorption schedule on clique trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.3 Junction Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.3.1 The running intersection property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.4 Constructing a Junction Tree for Singly-Connected Distributions . . . . . . . . . . . . . . . . 104 6.4.1 Moralisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4.2 Forming the clique graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4.3 Forming a junction tree from a clique graph . . . . . . . . . . . . . . . . . . . . . . . . 104 6.4.4 Assigning potentials to cliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Junction Trees for Multiply-Connected Distributions . . . . . . . . . . . . . . . . . . . . . . . 105 6.5.1 Triangulation algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.6 The Junction Tree Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.6.1 Remarks on the JTA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.6.2 Computing the normalisation constant of a distribution . . . . . . . . . . . . . . . . . 110 6.6.3 The marginal likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.6.4 Some small JTA examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.6.5 Shafer-Shenoy propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.7 Finding the Most Likely State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.8 Reabsorption : Converting a Junction Tree to a Directed Network . . . . . . . . . . . . . . . 114 6.9 The Need For Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.9.1 Bounded width junction trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.11 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.11.1 Utility routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7 Making Decisions 121 7.1 Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.1.1 Utility of money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.2 Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7.3 Extending Bayesian Networks for Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.3.1 Syntax of influence diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.4 Solving Influence Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4.1 Messages on an ID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4.2 Using a junction tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.5 Markov Decision Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.5.1 Maximising expected utility by message passing . . . . . . . . . . . . . . . . . . . . . . 134 7.5.2 Bellman’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.6 Temporally Unbounded MDPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.6.1 Value iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.6.2 Policy iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.6.3 A curse of dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.7 Variational Inference and Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.8 Financial Matters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.8.1 Options pricing and expected utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.8.2 Binomial options pricing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.8.3 Optimal investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.9 Further Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 DRAFT January 9, 2013 XIII
Page 1 and 2: Bayesian Reasoning and Machine Lear
Page 3 and 4: The data explosion Preface We live
Page 5 and 6: Part I: Inference in Probabilistic
Page 7 and 8: BRMLtoolbox The BRMLtoolbox is a li
Page 9 and 10: GMMem - Fit a mixture of Gaussian t
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Page 15 and 16: CONTENTS CONTENTS 9.5.2 Empirical i
Page 17 and 18: CONTENTS CONTENTS 15.3 High Dimensi
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Page 21 and 22: CONTENTS CONTENTS 25 Switching Line
Page 23 and 24: CONTENTS CONTENTS 29.1.6 Linear tra
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Page 28 and 29: Factor Graphs Graphical Models Undi
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Page 32 and 33: Probability Refresher The probabili
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Page 36 and 37: 1.1.2 Probability Tables Probabilis
Page 38 and 39: Probabilistic Reasoning where we us
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Page 46 and 47: and also p(x|y, z) = p(y|x, z)p(x|z
Page 48 and 49: Exercises 24 DRAFT January 9, 2013
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Page 54 and 55: node (a leaf node) which can be eli
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3.3 Belief Networks Definition 3.1
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x1 x3 (a) x2 x1 x3 (b) x2 x1 x3 (c)
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a b e A B C A B C A B C D A B C A B
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u x y z u x y w z u x y z u x y z u
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t1 y1 h (a) t2 y2 t1 y1 (b) t2 y2 B
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G D R (a) FD Resolution of the para
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where p(D|FD = ∅, G) ≡ p(D|pa (
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Battery Fuel Gauge Turn Over Start
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Exercises Exercise 3.15. A belief n
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Exercises 56 DRAFT January 9, 2013
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x1 x2 x4 (a) x3 x1 x2 x4 (b) x3 x1
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x1 x3 (a) x4 x3 (b) x2 x4 x1 x3 (c)
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Markov Networks 4-cycle x1 − x2
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Consider a model in which our desir
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Chain Graphical Models Each chain c
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Expressiveness of Graphical Models
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Consider the graph of the BN Expres
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2. By symmetry arguments, write dow
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Exercises The set of marginals cons
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discussed in more depth in section(
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Marginal Inference where v is a vec
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For consistency of interpretation,
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Marginal Inference This can be inte
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Other Forms of Inference defining m
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lists, see for example [72]. Other
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until the point Other Forms of Infe
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Inference in Multiply Connected Gra
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c a f d (a) marginal, say p(d). The
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Exercises demoShortestPath.m: The s
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Exercises 1. Explain mathematically
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a b c (a) d a, b, c b, c b, c, d (b
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A B C D ← 10 1 → 6 ← 2 → 3
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6.3.1 The running intersection prop
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dce d Constructing a Junction Tree
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To see this, consider the following
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a b c d e f g h i j k l (a) a b c d
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(a) 1 2 5 7 10 3 4 8 9 6 11 (b) 1 2
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• The new potential on (a, b) is
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Reabsorption : Converting a Junctio
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d1 d2 d3 d4 d5 s1 s2 s3 6.10 Summar
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Exercise 6.6. For the undirected gr
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P arty yes no Rain Rain (0.6) yes (
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P arty yes no (0.6) yes Rain (0.6)
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P arty Rain Uparty Partial ordering
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E UC The utilities are (a) P I UB E
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where D1 ≺ X1 ≺ D2 ≺ X2, and
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a b D1 U1 c d b, c, a b, c b, e, d,
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Markov Decision Processes time depe
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1 0 11 0 21 0 31 0 41 0 0 0 0 0 1 0
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Variational Inference and Planning
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Two possibilities case Financial Ma
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3 2.5 2 1.5 1 0.5 0 5 10 15 20 25 3
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Further Topics d1 d2 d3 Figure 7.13
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dx Ux a t e s x d l b dh Uh s = Smo
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Exercises We assume that we know wh
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The cost of playing the second stag
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Introduction to Part II In Part II
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CHAPTER 8 Statistics for Machine Le
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Distributions where the Jacobian ma
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Distributions Definition 8.10 (Empi
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Classical Distributions Definition
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Classical Distributions 5 4 3 2 1
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Classical Distributions x 3 1 0.5 0
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Multivariate Gaussian The multivari
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Exponential Family 8.4.3 Whitening
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Learning distributions In seeking t
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Properties of Maximum Likelihood Wh
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Learning a Gaussian (a) Prior (b) P
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Learning a Gaussian 8.8.3 Gauss-Gam
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Exercises Exercise 8.6. Using x T
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Exercises show that ∂ lim γ→0
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Exercises 2. Using equation (8.4.13
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Exercises 1. Show that the log prod
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CHAPTER 9 Learning as Inference In
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Learning as Inference In the coin t
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Bayesian methods and ML-II a s θ v
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Maximum Likelihood Training of Beli
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Bayesian Belief Network Training 9.
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Bayesian Belief Network Training Th
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Bayesian Belief Network Training No
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Structure learning Algorithm 9.1 PC
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Structure learning x y z w t (a) x
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Structure learning z1 θz z n x n y
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Structure learning Algorithm 9.3 Ch
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Maximum Likelihood for Undirected m
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Code • Provided we assume local a
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Exercises Exercise 9.3. A training
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CHAPTER 10 Naive Bayes So far we’
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Estimation using Maximum Likelihood
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Estimation using Maximum Likelihood
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Bayesian Naive Bayes For convenienc
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Exercises Practitioners typically c
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Exercises so that for example x2 =
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CHAPTER 11 Learning with Hidden Var
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Hidden Variables and Missing Data E
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Expectation Maximisation Algorithm
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Expectation Maximisation log p(v|θ
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Expectation Maximisation The first
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Expectation Maximisation Algorithm
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Extensions of EM log likelihood −
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A failure case for EM Stochastic EM
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Variational Bayes Algorithm 11.3 Va
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Variational Bayes θa a s c θc (a)
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Optimising the Likelihood by Gradie
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Exercises fuse assembly malfunction
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Exercises Exercise 11.9. A 2 × 2 p
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CHAPTER 12 Bayesian Model Selection
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Illustrations : coin tossing 8 6 4
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++ Occam’s Razor and Bayesian Com
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A continuous example : curve fittin
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Approximating the Model Likelihood
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Bayesian Hypothesis Testing for Out
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@@ @@ Exercises 1. Under the assump
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Exercises show that p(D|My→x) is
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Part III Machine Learning 285
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Clique decomp. Mixed membership Lat
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Train Test Styles of Learning Figur
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Supervised Learning be disastrous (
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X , C p(x, c) 〈L(c, c(x|θ))〉 p
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1.5 1 0.5 0 −0.5 −1 −1.5 −3
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Supervised Learning Based on zero-o
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θ c|h θh θ x|h cn xn hn N 13.3 B
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form p(c = 1|x) = 1 1 + exp −(ax
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K-Nearest Neighbours Figure 14.1: I
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A Probabilistic Interpretation of N
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Exercises many images of male faces
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3 2 1 0 −1 −2 0 2 4 4 2 0 −2
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2 1.5 1 0.5 0 −0.5 −1 −1.5
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Principal Components Analysis struc
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number of errors 12 10 8 6 4 2 0 0
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Latent Semantic Analysis Example 15
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(a) (b) PCA With Missing Data Figur
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(a) (b) Matrix Decomposition Method
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z x y M-step (a) x z y (b) Matrix D
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1 2 3 4 5 6 7 8 9 10 1 2 (a) 1 2 3
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factor 1 (Reinforcement Learning) 0
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Finding the reconstructions Canonic
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Exercises This approach is taken in
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3. Finding the eigenvalues and eige
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5 4 3 2 1 0 −1 −2 −3 −4 −
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Canonical Variates Between class Sc
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Exercises Example 16.1 (Using canon
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chirps per sec 110 105 100 95 90 85
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1 0.5 0 0 0.5 5 10 15 20 25 0 −0.
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17.2.3 Radial basis functions A pop
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17.3.1 Regression in the dual-space
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w Linear Parameter Models for Class
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0
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1.5 1 0.5 0 0.9 0.8 0.7 0.6 0.5 0.4
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2-Norm soft-margin w w origin ξ n
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2 1.5 1 0.5 0 −0.5 −1 −1.5
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17.8 Code demoCubicPoly.m: Demo of
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α w x n y n N β Regression With A
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Regression With Additive Gaussian N
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we obtain ∂ 1 log p(D|Γ) = ∂α
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18.1.7 Sparse linear models Regress
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18.2.1 Hyperparameter optimisation
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6 4 2 0 −2 −4 0.4 0.3 0.2 0.3 0
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1 0.5 0 −6 −4 −2 0 2 4 6 18.2
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8 6 4 2 0 −2 −4 −6 0.3 0.4 0.
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Exercises Exercise 18.6. This exerc
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θ y 1 y N y ∗ x 1 x N x ∗ (a)
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Gaussian Process Prediction The zer
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2 1.5 1 0.5 0 −0.5 −1 −1.5
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Covariance Functions For a stationa
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1.5 1 0.5 0 −0.5 −1 −1.5 −2
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c 1 c N c ∗ y 1 y N y ∗ x 1 x N
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Using these expressions in the Newt
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Gaussian Processes for Classificati
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d 1 log q(C|X ) = dθ 2 yTK −1 x,
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θh θ v|h h n v n N Expectation Ma
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θh θ vi|h M-step : p(v|h) h n vn
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1 2 3 4 Expectation Maximisation fo
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M-step : optimal mi Maximising equa
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(a) 1 iteration (b) 50 iterations (
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12 10 8 6 4 2 0 −2 −4 −6 −8
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8 6 4 2 0 −2 −4 −8 −6 −4
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z 1 v 1 θ (a) z N v N z 1 v 1 π
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π n z n w v n w α Wn θ N Mixed M
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(a) (b) Mixed Membership Models Fig
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1 0.8 0.6 0.4 0.2 0 −2 −1.5 −
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Mixed Membership Models 1000 Years
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Exercises Exercise 20.7. You wish t
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h1 h2 h3 v1 v2 v3 v4 v5 Probabilist
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21.2.1 Eigen-approach likelihood op
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Factor Analysis : Maximum Likelihoo
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¯x11 f1 G ¯x12 ¯x13 µ ¯x22 ori
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h1 h2 x1 x2 x∗ w1 w2 w∗ (a) M1
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h x y and integrating out the laten
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Exercises A popular alternative est
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Page 468 and 469:
δq xqs Q αs S The Rasch Model Fig
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competitor 10 20 30 40 50 60 70 80
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Exercises Exercise 22.4. An extensi
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Introduction to Part IV Natural org
Page 477 and 478:
CHAPTER 23 Discrete-State Markov Mo
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Markov Models h v1 v2 v3 v4 (a) Fig
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Markov Models For those less comfor
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Hidden Markov Models smoothing corr
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Hidden Markov Models This gives a r
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Hidden Markov Models (a) ‘Creaks
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Hidden Markov Models (a) (b) (c) Fi
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Learning HMMs M-step Assuming i.i.d
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Related Models c1 c2 c3 c4 h1 h2 h3
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Related Models x y1 y2 y3 Direction
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Applications determines the phoneme
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Exercises where Aij ≡ p(ht+1 = i|
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Exercises 1. First consider h1,...
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Exercises ++ Exercise 23.16 (Fuzzy
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CHAPTER 24 Continuous-state Markov
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Auto-Regressive Models 200 150 100
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Auto-Regressive Models a1 a2 a3 a4
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Latent Linear Dynamical Systems σ
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Inference messages exactly. In tran
Page 515 and 516:
Inference and covariance Ft ≡
Page 517 and 518:
Inference Algorithm 24.2 LDS Backwa
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Learning Linear Dynamical Systems y
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Learning Linear Dynamical Systems s
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Switching Auto-Regressive Models 10
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Code s1 s2 s3 s4 v1 v2 v3 v4 ˜v1
Page 527 and 528:
Exercises Exercise 24.5. Consider a
Page 529 and 530:
CHAPTER 25 Switching Linear Dynamic
Page 531 and 532:
Gaussian Sum Filtering than that of
Page 533 and 534:
Gaussian Sum Filtering t t+1 25.3.2
Page 535 and 536:
Gaussian Sum Smoothing filtered app
Page 537 and 538:
Gaussian Sum Smoothing Algorithm 25
Page 539 and 540:
Gaussian Sum Smoothing a b d 40 20
Page 541 and 542:
Reset Models s1 h1 v1 s2 h2 v2 s3 h
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Reset Models where g(a1, b1, a2, b2
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Exercises 11 10.5 10 9.5 0 50 100 1
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CHAPTER 26 Distributed Computation
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Learning Sequences v4(t) v3(t) v2(t
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Learning Sequences neuron number 10
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Learning Sequences (a) (b) Figure 2
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Tractable Continuous Latent Variabl
Page 557 and 558:
Neural Models Example 26.3 (Sequenc
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Neural Models 26.5.4 Leaky integrat
Page 561:
Part V Approximate Inference 537
Page 564 and 565:
Laplace (cont. variables) determini
Page 566 and 567:
Introduction For any sampling metho
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Algorithm 27.2 Rejection sampling t
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x1 x2 x3 x4 x5 x6 27.2 Ancestral Sa
Page 572 and 573:
x 2 x 1 Gibbs Sampling Figure 27.5:
Page 574 and 575:
3 2 1 0 −1 −2 −3 −4 −3
Page 576 and 577:
Algorithm 27.3 Metropolis-Hastings
Page 578 and 579:
Algorithm 27.4 Hybrid Monte Carlo s
Page 580 and 581:
Algorithm 27.5 Swendson-Wang sampli
Page 582 and 583:
Algorithm 27.6 Slice Sampling (univ
Page 584 and 585:
h1 h2 h3 h4 v1 v2 v3 v4 27.6.1 Sequ
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5 10 15 20 25 30 35 40 5 10 15 20 2
Page 588 and 589:
demoMetropolis.m: Demo of Metropoli
Page 590 and 591:
Page 592 and 593:
Properties of Kullback-Leibler Vari
Page 594 and 595:
Properties of Kullback-Leibler Vari
Page 596 and 597:
see fig(28.3), from which the poste
Page 598 and 599:
The variance is 2 zi − 〈zi〉
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Variational Bounding Using KL(q|p)
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Local and KL Variational Approximat
Page 604 and 605:
x y1 y2 y3 y4 Mutual Information Ma
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x1 x2 λx1→xi (xi) x3 λx2→xi (
Page 608 and 609:
Loopy Belief Propagation (so that t
Page 610 and 611:
Expectation Propagation ing an expo
Page 612 and 613:
which, on substituting in the updat
Page 614 and 615:
a b e f Lagrangian c d (a) a b e f
Page 616 and 617:
a+ b− s c+ d− t e− f+ a b (a)
Page 618 and 619:
Further Reading For a given α and
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1. ˜ f(x) ≥ ˜g(x), for x ≥ a
Page 622 and 623:
Exercise 28.13. Consider an approxi
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Page 626 and 627:
e e* a Linear Algebra a α e Figure
Page 628 and 629:
Linear Algebra A square matrix A is
Page 630 and 631:
Linear Algebra Definition 29.12 (Bl
Page 632 and 633:
Definition 29.17 (Quadratic form).
Page 634 and 635:
Inequalities Definition 29.20 (Chai
Page 636 and 637:
29.5.1 Gradient descent with fixed
Page 638 and 639:
29.5.5 The conjugate vectors algori
Page 640 and 641:
Algorithm 29.2 Quasi-Newton for min
Page 642 and 643:
Constrained Optimisation using Lagr
Page 644 and 645:
BIBLIOGRAPHY BIBLIOGRAPHY [16] D. B
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BIBLIOGRAPHY BIBLIOGRAPHY [63] S. C
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BIBLIOGRAPHY BIBLIOGRAPHY [111] A.
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BIBLIOGRAPHY BIBLIOGRAPHY [159] F.
Page 652 and 653:
BIBLIOGRAPHY BIBLIOGRAPHY [206] M.
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BIBLIOGRAPHY BIBLIOGRAPHY [251] F.
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BIBLIOGRAPHY BIBLIOGRAPHY [299] M.
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BIBLIOGRAPHY BIBLIOGRAPHY 634 DRAFT
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INDEX INDEX Bellman’s equation, 1
Page 662 and 663:
INDEX INDEX conditioning, 170 conju
Page 664 and 665:
INDEX INDEX filtering, 459 input-ou
Page 666 and 667:
INDEX INDEX symmetrising updates, 4
Page 668 and 669:
INDEX INDEX conjugate vectors algor
Page 670:
INDEX INDEX changepoint model, 516
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Bayesian Reasoning and Machine Learning

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