Bayesian Reasoning and Machine Learning

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CONTENTS CONTENTS 23.2.7 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 23.2.8 Self localisation and kidnapped robots . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 23.2.9 Natural language models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 23.3 Learning HMMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 23.3.1 EM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 23.3.2 Mixture emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 23.3.3 The HMM-GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 23.3.4 Discriminative training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 23.4 Related Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 23.4.1 Explicit duration model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 23.4.2 Input-Output HMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 23.4.3 Linear chain CRFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 23.4.4 Dynamic Bayesian networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 23.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 23.5.1 Object tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 23.5.2 Automatic speech recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 23.5.3 Bioinformatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 23.5.4 Part-of-speech tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 23.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 23.7 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 23.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 24 Continuous-state Markov Models 481 24.1 Observed Linear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 24.1.1 Stationary distribution with noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 24.2 Auto-Regressive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 24.2.1 Training an AR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 24.2.2 AR model as an OLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 24.2.3 Time-varying AR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 24.2.4 Time-varying variance AR models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 24.3 Latent Linear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 24.4 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 24.4.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 24.4.2 Smoothing : Rauch-Tung-Striebel correction method . . . . . . . . . . . . . . . . . . . 492 24.4.3 The likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 24.4.4 Most likely state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 24.4.5 Time independence and Riccati equations . . . . . . . . . . . . . . . . . . . . . . . . . 494 24.5 Learning Linear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 24.5.1 Identifiability issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 24.5.2 EM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 24.5.3 Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 24.5.4 Structured LDSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 24.5.5 Bayesian LDSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 24.6 Switching Auto-Regressive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 24.6.1 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 24.6.2 Maximum likelihood learning using EM . . . . . . . . . . . . . . . . . . . . . . . . . . 499 24.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 24.8 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 24.8.1 Autoregressive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 24.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 XX DRAFT January 9, 2013
CONTENTS CONTENTS 25 Switching Linear Dynamical Systems 505 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 25.2 The Switching LDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 25.2.1 Exact inference is computationally intractable . . . . . . . . . . . . . . . . . . . . . . . 506 25.3 Gaussian Sum Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 25.3.1 Continuous filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 25.3.2 Discrete filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 25.3.3 The likelihood p(v1:T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 25.3.4 Collapsing Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 25.3.5 Relation to other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 25.4 Gaussian Sum Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 25.4.1 Continuous smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 25.4.2 Discrete smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 25.4.3 Collapsing the mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 25.4.4 Using mixtures in smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 25.4.5 Relation to other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 25.5 Reset Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 25.5.1 A Poisson reset model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 25.5.2 Reset-HMM-LDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 25.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 25.7 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 25.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 26 Distributed Computation 523 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 26.2 Stochastic Hopfield Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 26.3 Learning Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 26.3.1 A single sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 26.3.2 Multiple sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 26.3.3 Boolean networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 26.3.4 Sequence disambiguation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 26.4 Tractable Continuous Latent Variable Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 26.4.1 Deterministic latent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 26.4.2 An augmented Hopfield network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 26.5 Neural Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 26.5.1 Stochastically spiking neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 26.5.2 Hopfield membrane potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 26.5.3 Dynamic synapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 26.5.4 Leaky integrate and fire models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 26.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 26.7 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 26.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 V Approximate Inference 537 27 Sampling 541 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 27.1.1 Univariate sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 27.1.2 Rejection sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 27.1.3 Multivariate sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 27.2 Ancestral Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 27.2.1 Dealing with evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 27.2.2 Perfect sampling for a Markov network . . . . . . . . . . . . . . . . . . . . . . . . . . 547 27.3 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 27.3.1 Gibbs sampling as a Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 DRAFT January 9, 2013 XXI
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
Page 11 and 12: Contents Front Matter I Notation Li
Page 13 and 14: CONTENTS CONTENTS 5.6.3 Bucket elim
Page 15 and 16: CONTENTS CONTENTS 9.5.2 Empirical i
Page 17 and 18: CONTENTS CONTENTS 15.3 High Dimensi
Page 19: CONTENTS CONTENTS 20.3.7 Semi-super
Page 23 and 24: CONTENTS CONTENTS 29.1.6 Linear tra
Page 25: Part I Inference in Probabilistic M
Page 28 and 29: Factor Graphs Graphical Models Undi
Page 30 and 31: 6 DRAFT January 9, 2013
Page 32 and 33: Probability Refresher The probabili
Page 34 and 35: Probability Refresher conditioning
Page 36 and 37: 1.1.2 Probability Tables Probabilis
Page 38 and 39: Probabilistic Reasoning where we us
Page 40 and 41: Then p(A = 0, C = 0) = p(A = 0, B,
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Page 44 and 45: 1.5 Code Code The BRMLtoolbox code
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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Page 60 and 61: Uncertain and Unreliable Evidence k
Page 62 and 63: 3.3 Belief Networks Definition 3.1
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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
Page 414 and 415:
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
Page 460 and 461:
h1 h2 x1 x2 x∗ w1 w2 w∗ (a) M1
Page 462 and 463:
h x y and integrating out the laten
Page 464 and 465:
Exercises A popular alternative est
Page 466 and 467:
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
Page 472 and 473:
Exercises Exercise 22.4. An extensi
Page 475 and 476:
Introduction to Part IV Natural org
Page 477 and 478:
CHAPTER 23 Discrete-State Markov Mo
Page 479 and 480:
Markov Models h v1 v2 v3 v4 (a) Fig
Page 481 and 482:
Markov Models For those less comfor
Page 483 and 484:
Hidden Markov Models smoothing corr
Page 485 and 486:
Hidden Markov Models This gives a r
Page 487 and 488:
Hidden Markov Models (a) ‘Creaks
Page 489 and 490:
Hidden Markov Models (a) (b) (c) Fi
Page 491 and 492:
Learning HMMs M-step Assuming i.i.d
Page 493 and 494:
Related Models c1 c2 c3 c4 h1 h2 h3
Page 495 and 496:
Related Models x y1 y2 y3 Direction
Page 497 and 498:
Applications determines the phoneme
Page 499 and 500:
Exercises where Aij ≡ p(ht+1 = i|
Page 501 and 502:
Exercises 1. First consider h1,...
Page 503 and 504:
Exercises ++ Exercise 23.16 (Fuzzy
Page 505 and 506:
CHAPTER 24 Continuous-state Markov
Page 507 and 508:
Auto-Regressive Models 200 150 100
Page 509 and 510:
Auto-Regressive Models a1 a2 a3 a4
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Latent Linear Dynamical Systems σ
Page 513 and 514:
Inference messages exactly. In tran
Page 515 and 516:
Inference and covariance Ft ≡
Page 517 and 518:
Inference Algorithm 24.2 LDS Backwa
Page 519 and 520:
Learning Linear Dynamical Systems y
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Learning Linear Dynamical Systems s
Page 523 and 524:
Switching Auto-Regressive Models 10
Page 525 and 526:
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
Page 543 and 544:
Reset Models where g(a1, b1, a2, b2
Page 545 and 546:
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
Page 555 and 556:
Tractable Continuous Latent Variabl
Page 557 and 558:
Neural Models Example 26.3 (Sequenc
Page 559 and 560:
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
Page 568 and 569:
Algorithm 27.2 Rejection sampling t
Page 570 and 571:
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
Page 586 and 587:
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〉
Page 600 and 601:
Variational Bounding Using KL(q|p)
Page 602 and 603:
Local and KL Variational Approximat
Page 604 and 605:
x y1 y2 y3 y4 Mutual Information Ma
Page 606 and 607:
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
Page 624 and 625:
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
Page 646 and 647:
BIBLIOGRAPHY BIBLIOGRAPHY [63] S. C
Page 648 and 649:
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.
Page 658 and 659:
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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