Bayesian Reasoning and Machine Learning

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1.5 Code Code The BRMLtoolbox code accompanying this book is intended to give the reader some insight into representing discrete probability tables and performing simple inference. We provide here only the briefest of descriptions of the code and the reader is encouraged to experiment with the demos to understand better the routines and their purposes. 1.5.1 Basic Probability code At the simplest level, we only need two basic routines. One for multiplying probability tables together (called potentials in the code), and one for summing a probability table. Potentials are represented using a structure. For example, in the code corresponding to the Inspector Clouseau example demoClouseau.m, we define a probability table as >> pot(1) ans = variables: [1 3 2] table: [2x2x2 double] This says that the potential depends on the variables 1, 3, 2 and the entries are stored in the array given by the table field. The size of the array informs how many states each variable takes in the order given by variables. The order in which the variables are defined in a potential is irrelevant provided that one indexes the array consistently. A routine that can help with setting table entries is setstate.m. For example, >> pot(1) = setstate(pot(1),[2 1 3],[2 1 1],0.3) means that for potential 1, the table entry for variable 2 being in state 2, variable 1 being in state 1 and variable 3 being in state 1 should be set to value 0.3. The philosophy of the code is to keep the information required to perform computations to a minimum. Additional information about the labels of variables and their domains can be useful to interpret results, but is not actually required to carry out computations. One may also specify the name and domain of each variable, for example >>variable(3) ans = domain: {’murderer’ ’not murderer’} name: ’butler’ The variable name and domain information in the Clouseau example is stored in the structure variable, which can be helpful to display the potential table: >> disptable(pot(1),variable); knife = used maid = murderer butler = murderer 0.100000 knife = not used maid = murderer butler = murderer 0.900000 knife = used maid = not murderer butler = murderer 0.600000 knife = not used maid = not murderer butler = murderer 0.400000 knife = used maid = murderer butler = not murderer 0.200000 knife = not used maid = murderer butler = not murderer 0.800000 knife = used maid = not murderer butler = not murderer 0.300000 knife = not used maid = not murderer butler = not murderer 0.700000 Multiplying Potentials In order to multiply potentials, (as for arrays) the tables of each potential must be dimensionally consistent – that is the number of states of variable i must be the same for all potentials. This can be checked using potvariables.m. This consistency is also required for other basic operations such as summing potentials. multpots.m: Multiplying two or more potentials divpots.m: Dividing a potential by another 20 DRAFT January 9, 2013
Exercises Summing a Potential sumpot.m: Sum (marginalise) a potential over a set of variables sumpots.m: Sum a set of potentials together Making a conditional Potential condpot.m: Make a potential conditioned on variables Setting a Potential setpot.m: Set variables in a potential to given states setevpot.m: Set variables in a potential to given states and return also an identity potential on the given states The philosophy of BRMLtoolbox is that all information about variables is local and is read off from a potential. Using setevpot.m enables one to set variables in a state whilst maintaining information about the number of states of a variable. Maximising a Potential maxpot.m: Maximise a potential over a set of variables See also maxNarray.m and maxNpot.m which return the N-highest values and associated states. Other potential utilities setstate.m: Set a potential state to a given value table.m: Return a table from a potential whichpot.m: Return potentials which contain a set of variables potvariables.m: Variables and their number of states in a set of potentials orderpotfields.m: Order the fields of a potential structure uniquepots.m: Merge redundant potentials by multiplication and return only unique ones numstates.m: Number of states of a variable in a domain squeezepots.m: Find unique potentials and rename the variables 1,2,... normpot.m: Normalise a potential to form a distribution 1.5.2 General utilities condp.m: Return a table p(x|y) from p(x, y) condexp.m: Form a conditional distribution from a log value logsumexp.m: Compute the log of a sum of exponentials in a numerically precise way normp.m: Return a normalised table from an unnormalised table assign.m: Assign values to multiple variables maxarray.m: Maximize a multi-dimensional array over a subset 1.5.3 An example The following code highlights the use of the above routines in solving the Inspector Clouseau, example(1.3), and the reader is invited to examine the code to become familiar with how to numerically represent probability tables. demoClouseau.m: Solving the Inspector Clouseau example 1.6 Exercises Exercise 1.1. Prove p(x, y|z) = p(x|z)p(y|x, z) (1.6.1) DRAFT January 9, 2013 21
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 and 20: CONTENTS CONTENTS 20.3.7 Semi-super
Page 21 and 22: CONTENTS CONTENTS 25 Switching Line
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 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
Page 50 and 51: Graphs and each edge (Ak−1, Ak),
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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
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Page 64 and 65: x1 x3 (a) x2 x1 x3 (b) x2 x1 x3 (c)
Page 66 and 67: a b e A B C A B C A B C D A B C A B
Page 68 and 69: u x y z u x y w z u x y z u x y z u
Page 70 and 71: t1 y1 h (a) t2 y2 t1 y1 (b) t2 y2 B
Page 72 and 73: G D R (a) FD Resolution of the para
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Page 76 and 77: Battery Fuel Gauge Turn Over Start
Page 78 and 79: Exercises Exercise 3.15. A belief n
Page 80 and 81: Exercises 56 DRAFT January 9, 2013
Page 82 and 83: x1 x2 x4 (a) x3 x1 x2 x4 (b) x3 x1
Page 84 and 85: x1 x3 (a) x4 x3 (b) x2 x4 x1 x3 (c)
Page 86 and 87: Markov Networks 4-cycle x1 − x2
Page 88 and 89: Consider a model in which our desir
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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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Exercises 120 DRAFT January 9, 2013
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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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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
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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
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Inference and covariance Ft ≡
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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
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Exercises Exercise 24.5. Consider a
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CHAPTER 25 Switching Linear Dynamic
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Gaussian Sum Filtering than that of
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Gaussian Sum Filtering t t+1 25.3.2
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Gaussian Sum Smoothing filtered app
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Gaussian Sum Smoothing Algorithm 25
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Gaussian Sum Smoothing a b d 40 20
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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
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Neural Models Example 26.3 (Sequenc
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Neural Models 26.5.4 Leaky integrat
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Part V Approximate Inference 537
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Laplace (cont. variables) determini
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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
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x 2 x 1 Gibbs Sampling Figure 27.5:
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3 2 1 0 −1 −2 −3 −4 −3
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Algorithm 27.3 Metropolis-Hastings
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Algorithm 27.4 Hybrid Monte Carlo s
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Algorithm 27.5 Swendson-Wang sampli
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Algorithm 27.6 Slice Sampling (univ
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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
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demoMetropolis.m: Demo of Metropoli
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Properties of Kullback-Leibler Vari
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Properties of Kullback-Leibler Vari
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see fig(28.3), from which the poste
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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
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x y1 y2 y3 y4 Mutual Information Ma
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x1 x2 λx1→xi (xi) x3 λx2→xi (
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Loopy Belief Propagation (so that t
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Expectation Propagation ing an expo
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which, on substituting in the updat
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a b e f Lagrangian c d (a) a b e f
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a+ b− s c+ d− t e− f+ a b (a)
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Further Reading For a given α and
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1. ˜ f(x) ≥ ˜g(x), for x ≥ a
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Exercise 28.13. Consider an approxi
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e e* a Linear Algebra a α e Figure
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Linear Algebra A square matrix A is
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Linear Algebra Definition 29.12 (Bl
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Definition 29.17 (Quadratic form).
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Inequalities Definition 29.20 (Chai
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29.5.1 Gradient descent with fixed
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29.5.5 The conjugate vectors algori
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Algorithm 29.2 Quasi-Newton for min
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Constrained Optimisation using Lagr
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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.
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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
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INDEX INDEX conditioning, 170 conju
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INDEX INDEX filtering, 459 input-ou
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INDEX INDEX symmetrising updates, 4
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INDEX INDEX conjugate vectors algor
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INDEX INDEX changepoint model, 516
show all

Bayesian Reasoning and Machine Learning

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