books.google.de - Machine learning methods extract value from vast data sets quickly and with modest resources. They are established tools in a wide range of industrial applications, including search engines, DNA sequencing, stock market analysis, and robot locomotion, and their use is spreading rapidly. People who know...https://books.google.de/books/about/Bayesian_Reasoning_and_Machine_Learning.html?id=yxZtddB_Ob0C&utm_source=gb-gplus-shareBayesian Reasoning and Machine Learning
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David Barber. inference \ v=s \_\1\_e-:m\r\T_,\ '“ “' '85' t data BAYLESIAN hm '1.M\f
€ R'€A4$f?_5i\"N<3 \ \ NATL AEZ 1' 1 'LE/WING David Barber Extracting value
from vast amounts of data presents a major. Front Cover.
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David Barber. Extracting value from vast amounts of data presents a major
challenge to all those working in computer science and related fields. Machine
learning technology is already used to help with this task in a wide range of
industrial ...
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David Barber. Machine Learning David Barber University College London
cambridge university press Cambridge, New York, Melbourne, Madrid, Cape
Town,. Bayesian Reasoning and.
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David Barber. cambridge university press Cambridge, New York, Melbourne,
Madrid, Cape Town, Singapore, S ̃ao Paulo, Delhi, Tokyo, Mexico City
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK
Published in ...
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David Barber. 8 9 165 199 10 11 II 284 III 13 14 15. Learning in probabilistic
models Statistics for machine learning 8.1 Representing data 8.1.1 Categorical
8.1.2 Ordinal 8.1.3 Numerical 8.2 Distributions 8.2.1 The Kullback–Leibler
divergence ...
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David Barber. 284 III 13 14 15 15.3 15.4 15.5 15.6 15.7 15.8 High-dimensional
data 15.3.1 Eigen-decomposition ... E-step A failure case for EM Variational
Bayes 11.5.1 EM is a special case of variational Bayes 11.5.2 An example: VB for
the ...
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David Barber. 15.3 15.4 15.5 15.6 15.7 15.8 High-dimensional data 15.3.1 Eigen-
decomposition for N<D 15.3.2 PCA via singular value decomposition Latent
semantic analysis 15.4.1 Information retrieval PCA with missing data 15.5.1
Finding ...
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David Barber. 21 22 25.5 25.6 25.7 25.8 26 Distributed computation 26.1 26.2
26.3. 19 Gaussian processes 19.1 19.2 ... 20.3 412 Non-parametric prediction
19.1.1 From parametric to non-parametric 19.1.2 From Bayesian linear models to
...
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David Barber. 22.3 Summary 22.4 Code 22.5 Exercises IV Dynamical models 23
Discrete-state Markov models 489 23.1 Markov models 23.1.1 Equilibrium and
stationary distribution of a Markov chain 23.1.2 Fitting Markov models 23.1.3 ...
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David Barber. The. data. explosion. We live in a world that is rich in data, ever
increasing in scale. This data comes from many different sources in science (
bioinformatics, astronomy, physics, environmental monitoring) and commerce ...
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David Barber. raphica l m o d e l s c o u r s e P r o b a b i l i s t i c m a c h i n e
learnin g c o u r s e P r o b a b i l i s t i c m odellin g c o u r s e A pproxi m at
e i n f e r e n c e s h o r t c o u r s e T i m e s e r i e s s h o r t c o u r s e G
Accompanying ...
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The BRMLtoolbox along with an electronic version of the book is available from
www.cs.ucl.ac.uk/staff/D.Barber/brml Instructors seeking solutions to the
exercises can find information at www.cambridge.org/brml, along with additional
teaching ...
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David Barber. numstates orderpot orderpotfields potsample potscontainingonly
potvariables setevpot setpot setstate squeezepots sumpot sumpotID sumpots
table ungrouppot uniquepots whichpot - Number of states of the variables in a ...
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David Barber. cca covfnGE FA GMMem GPclass GPreg HebbML HMMbackward
HMMbackwardSAR HMMem HMMforward HMMforwardSAR HMMgamma
yHMMsmooth HMMsmoothSAR HMMviterbi kernel Kmeans LDSbackward ...
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David Barber. 1.1 We have intuition about how uncertainty works in simple cases.
To reach sensible conclusions in ... In this chapter we review basic concepts in
probability – in particular, conditional probability and Bayes' rule, the workhorses
...
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David Barber. is in state x. Conversely, p(x = x) = 0 means that we are certain x is
not in state x. Values between 0 and 1 represent the degree of certainty of state
occupancy. The summation of the probability over all the states is 1: ∑ x∈dom(x)
...
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David Barber ... Reasoning (inference) is then performed by introducing evidence
that sets variables in known ... The rules of probability, combined with Bayes' rule
make for a complete reasoning system, one which includes traditional ...
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David Barber. Example 1.6 Aristotle: Inverse Modus Ponens According to Logic,
from the statement: 'If A is true then B is true', one may deduce that 'If B is false
then A is false'. To see how this fits in with a probabilistic reasoning system we
can ...
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David Barber. Example 1.8 Larry Larry is typically late for school. If Larry is late,
we denote this with L : late, otherwise, L : not late. When his mother asks whether
or not he was late for school he never admits to being late. The response Larry ...
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David Barber. 1.3 Given that RS = late and RL = not late, what is the probability
that Larry was late? Using Bayes' rule, we have p(L = late|RL = not late,R S = late
) = 1 Zp(RS = late|L = late)p(RL = not late|L = late)p(L = late) where the ...
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David Barber ... r The standard rules of probability are a consistent, logical way to
reason with uncertainty. r Bayes' rule mathematically encodes the process of
inference. A useful introduction to probability is given in [292]. The interpretation
of ...
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David Barber. 3.1 We can now make a first connection between probability and
graph theory. A belief network introduces structure into a probabilistic model by
using graphs to represent independence assumptions among the variables.
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David Barber. 3.3 Using this modified model, we can now use Jeffrey's rule to
compute the model conditioned on the evidence p(B)p(A|B)p(G|A)p(H|A) p(B,A|H,
G) = ∑ A,B p(B)p(A|B)p(G|A)p(H|A) . (3.2.11) We now include the uncertain ...
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David Barber. x y z u ⇒ x y z u If z is a collider (bottom path) keep undirected
links between the neighbours of the collider. x y w z u ⇒ x y w z u If z is a
descendant of a collider, this could induce dependence so we retain the links (
making them ...
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David Barber. 3.3.5 Remark 3.5 (Bayes ball) The Bayes ball algorithm [258]
provides a linear time complexity algorithm which given a set of nodes X and Z
determines the set of nodes Y such that X ⊥⊥Y|Z. Y is called the set of irrelevant
...
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David Barber. 6.2. From the definition of conditional probability, we can reexpress
this as p(a,b) p(b) p(b, c) p(c) p(c, d) p(d) p(a,b)p(b, c)p(c, d) p(b)p(c) A useful
insight is that the distribution can therefore be written as a product of marginal ...
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David Barber ... An influence diagram is a Bayesian network with additional
decision nodes and utility nodes [149, 161, 175]. The decision nodes have no
associated distribution and the utility nodes are deterministic functions of their
parents.
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David Barber. 8.5.1 8.6 Conjugate priors For an exponential family likelihood p(x|
θ) = h(x)exp ( θTT(x) − ψ (θ) ) (8.5.6) and prior with hyperparameters α,γ, p(θ|α,γ)
∝ exp ( θTα − γψ (θ) ) (8.5.7) the posterior is p(θ|x,α,γ) ∝ p(x|θ)p(θ|α,γ) ∝ exp ...
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David Barber. define models for which the resulting computational difficulties are
minimal, or in finding good approximations ... We first reiterate some of the basic
ground covered in Section 1.3, in which the Bayesian and maximum likelihood ...
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David Barber. 8.8.2 Optimal )1. Taking the partial derivative with respect to the
vector p. we obtain the vector derivative 1v v,,L(,t, 2) = Z 2*' (X" - ]L). (8.8.2) n:l
Equating to zero gives that at the optimum of the log likelihood, N Z zclx" = 1v,t>:*
1 ...
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David Barber. 8.9 8.10 8.11 8.1 is therefore the conjugate prior for a Gaussian
with unknown mean μ and precision λ. The posterior for this prior is a Gauss-
gamma distribution with parameters ̃b ̃a,1 ̃aλ ) Gam(λ|α+N/2, ̃β). (8.8.28) p(μ,λ|X,
μ0,α,β ...
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David Barber ... p(θ|θ) with hyperparameter θ. This is depicted in Fig. 9.5(b).
Learning then 9.1.4 9.3 c a s (a) cn an sn θ c 9.2 Bayesian methods and ML-II
203 9.1.4 Decisions based on continuous intervals 9.2 Bayesian methods and
ML-II.
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David Barber. 9.4 9.4.1 Bayesian belief network training ... An alternative to
maximum likelihood training of a BN is to use a Bayesian approach in which we
maintain a distribution over parameters. We continue with the Asbestos, Smoking
, ...
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David Barber. The prior parameters αc (a,s) are called hyperparameters. A
complete ignorance prior would correspond ... It is instructive to examine this
Bayesian solution under various conditions: No data limit N → 0 In the limit of no
data, the ...
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David Barber. Algorithm 9.2 Skeleton orientation algorithm (returns a DAG). 1:
Unmarried collider: Examine all undirected links x—z—y. lfz 6? 8,}, setx ... Figure
9.12 Bayesian conditional independence test using Dirichlet priors on the tables.
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David Barber. 9.5.3 z 1 x y z 2 Figure 9.13 Conditional independence test of x⊥
⊥y |z 1 , z2 with x, y, z1, z2 having 3,2, 4, ... the Bayesian conditional
independence test correctly states that the variables are conditionally
independent 74% of ...
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David Barber. Figure 9.14 Learning the structure of a Bayesian network. (a) The
correct structure in which all variables are binary. The ancestral order is x2, x1, x5
, x4, x3, x8, x7, x6. The dataset is formed from 1000 samples from this network.
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David Barber. 9.8.3 9.9 9.1 is false. We then measure the fraction of experiments
for which the Bayes test correctly decides x⊥⊥y|z. We also measure the fraction
of experiments for which the Mutual Information (MI) test correctly decides ...
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David Barber. 10.1 So far we've discussed methods in some generality without
touching much on how we might use the methods in a practical setting. Here we
discuss one of the simplest methods that is widely used in practice to classify
data.
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David Barber. 10.2 case. Also, the attributes xi are often taken to be binary, as we
shall do initially below as well. The extension to more than two attribute states, or
continuous attributes is straightforward. Example 10.1 EZsurvey.org partitions ...
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David Barber. Classification boundary We classify a novel input x∗ as class 1 if p
(c = 1|x∗) > p(c = 0|x∗). (10.2.8) Using Bayes' rule and writing the log of the
above expression, this is equivalent to logp(x∗|c = 1) + logp(c = 1) − logp(x∗) ...
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David Barber. 0 1 1 1 0 0 1 1 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 0 Figure 10.2 (a)
English tastes for 6 1 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 1 0 1 1 1 1 0
peopleoverattributes(shortbread,lager,whiskey,porridge,football).Eachcolumn
represents the ...
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David Barber ... A naive Bayes model specifies a distribution of these number of
occurrences p(xi |c), where xi is the count of the number of times word i appears
in documents of type c. One can achieve this using either a multistate ...
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David Barber. cn xni θi,c n =1: N θc c =1: C i =1: D Figure 10.3 Bayesian naive
Bayes with a factorised prior on the class conditional attribute probabilities p(xi =
s|c). For simplicity we assume that the class probability θc ≡ p(c) is learned with
...
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David Barber. 10.4 Classification The posterior class distribution for a novel input
x* is given by ... (10.3.15) Example 10.3 Bayesian naive Bayes Repeating the
previous analysis for the 'Are they Scottish?' data from Example 10.2, the ...
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David Barber. 10.4.1 10.5 10.6 c x 1 x 2 x 3 x 4 Figure 10.4 Tree Augmented
Naive (TAN) Bayes. Each variable xi has at most one parent. The maximum
likelihood optimal TAN structure is computed using a modified Chow–Liu
algorithm in ...
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David Barber ... NaiveBayesDirichletTest.m: Naive Bayes testing with Bayesian
Dirichlet demoNaiveBayes.m: Demo of naive Bayes ... Using naive Bayes trained
with maximum likelihood, what is the probability that she is younger than 60?
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David Barber. 10.4 10.5 10.6 Training data from sport documents and from
politics documents is represented below in ... Using a maximum likelihood naive
Bayes classifier, what is the probability that the document x = (1,0,0,1,1,1,1,0)is
about ...
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David Barber. 11.5 @ I Figure 11.5 (a) Generic form ofa model with hidden N
variables. (b) A factorised posterior approximation uses in variational Bayes. e 0
~ (9) (I9) _ a similar slowing down of parameter updates can occur when the term
...
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David Barber. Algorithm 11.3 Variational Bayes. : end while 1: t: O > Iteration
counter 2: Choose an initial distribution qo (6). > lnitialisation 3: while 6 not
converged (or likelihood bound not converged) do 42 t<— t+ 1 5: q,(H) = arg
minqmy ...
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David Barber. 11.5.1 Algorithm 11.4 Variational Bayes (i.i.d. data). 1: t: 0 >
Iteration counter 2: Choose an initial ... EM is a special case of variational Bayes
If we wish to find a summary of the parameter posterior corresponding to only the
most ...
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David Barber. 11.5.2 where θ∗ is the single optimal value of the parameter. If we
plug this assumption into Equation (11.5.4) we obtain the bound logp(v|θ∗) ≥ − 〈
logq(h)〉q(h) + 〈logp(v,h,θ ∗)〉q(h) + const. (11.5.14) The M-step is then given ...
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David Barber. 11.7 11.8 For i.i.d. data, the log likelihood on the visible variables
is (assuming discrete v and h) L(θ) = ∑n ( log ∑ h expφ(vn,h|θ) − log ∑ h,v expφ
(v,h|θ) ) (11.6.5) which has gradient ∂ ∂θ L= ∑ n ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 〈 ∂∂θφ(vn,h|θ)
〉 ...
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David Barber. 12 Bayesian model selection 12.1 So far we've mostly used Bayes'
rule for inference at the parameter level. Applied at the model level, Bayes' rule
gives a method for evaluating competing models. This provides an alternative to
...
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David Barber. 12.3 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0 2 4 6 8 (a) (
b) Figure 12.2 Probability density priors on the probability ofa Head p(θ). (a) For a
fair coin we choose p(θ|Mfair) = B (θ|50, 50). (b) For a biased we choose ...
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David Barber. 12.4 0 0.2 0 0 0.1 0.2 0 0.1 0 0.10.2 0 0.1 0.2 0 0 0.2 9 10 11 12 13
1415 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 0 0.1 123 4 5 6 78 Figure
12.3 The likelihood of the total dice score, p(t|n) for n = 1 (top) to n = 5 (bottom)
die.
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David Barber. 12.5 12.5.1 −10 −5 0 5 10 −3 −2 −1 0 1 2 3 12345678910 0 0.2 0.4
0.6 0.8 1 −10 −5 0 5 10 −3 −2 −1 0 1 2 3 (a) (b) (c) Figure 12.6 (a) The data
generated with additive Gaussian noise σ = 0.5 from a K = 5 component model.
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David Barber. 12.5.2. 12.6. 12.6 Bayesian hypothesis testing for outcome
analysis For data D = { x1,...,xN } that is i.i.d. generated the above specialises to p
(D|M) = ∫ p(θ|M)N∏n=1p(xn|θ,M)dθ. (12.5.5) In this case Laplace's method
computes ...
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David Barber. 12.6.1 performing differently. The techniques we discuss are quite
general, but we phrase them in terms of ... For techniques which are based on
Bayesian classifiers there will always be, in principle, a direct way to estimate the
...
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David Barber. 12.6.3 The prior hyperparameter u controls how strongly the mass
of the distribution is pushed to the corners of the simplex, see Fig. 8.6. Setting uq
= 1 for all q corresponds to a uniform prior. The likelihood of observing oa is ...
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David Barber. 12.6.4 Below we discuss some further examples for the Hindep
versus Hsame test. As above, the only ... The posterior Bayes' factor equation (
12.6.15) is 20.7 – strong evidence in favour of the two classifiers being different. r
...
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David Barber. Using a ,1 b p(0A Iea) : Qgwrrw _ 6a)n1n(0rre<t , p(0B : Qgwrrerl _
6b)ni11mne<1 and Beta distribution priors p(t9a) : B (t9,,|u1, 142) , 12(65): B (6b|
u1, M2) (12.6.28) then one may readily show, using the Beta function B(x, ...
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David Barber. 12.7 30 (b) 25200123456789 (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.9 1 0 5 10 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 12.8 Two classifiers A
and B and their posterior distributions of the probability that they classify ...
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David Barber. 12.8. 12.9. 12.1 12.2. Code. demoBayesErrorAnalysis.m: Demo for
Bayesian error analysis betaXbiggerY.m: p(x>y) forx ∼ B (x|a,b), y ∼ B (y|c, d).
Exercises. Write a program to implement the fair/biased coin tossing model ...
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David Barber. X,C p(x, c|θ) Figure 13.6 Bayesian decision approach. A model p(x
,c|θ) is fitted to the data. After learning the optimal model parameters θ, we
compute p(c|x, θ). For a novel x∗, the distribution of the assumed 'truth' is p(c|x∗,
θ).
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David Barber. 13.3 θc|h θh θx|h N cnxnhn as the distance between eyes, width of
mouth, etc. may make finding a classifier easier. In practice data is also often
preprocessed to remove noise, centre an image etc. Figure 13.9 A strategy for ...
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David Barber. decision function parameters are set based on the task of making
decisions. On the other hand, the Bayesian approach attempts to learn
meaningful p(c, x) without regard to its ultimate use as part of a larger decision
process.
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David Barber. 13.4 13.5 13.1 13.2 13.3 13.4 Both approaches are heavily used in
practice and which is to be preferred depends very much on the ... Whilst the
Bayesian approach appears formally optimal, it is prone to model mis-
specification.
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David Barber. Table 15.1 Highest ranked documents according to p(c|z). The
factor topic labels are manual assignments based on similarity to the Cora topics.
Reproduced from [66]. factor 1 (Reinforcement Learning) 0.0108 Learning to ...
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David Barber. 18 Bayesian linear models 18.1 The previous chapter discussed
the use of linear models in classification and regression. In this chapter we
discuss using priors on the parameters and the resulting posterior distribution
over ...
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David Barber. 18.1.1 wαβxnynN Using the Gaussian noise assumption, and for
convenience defining β = 1/σ2, this gives Figure 18.1 Belief network
representation of a Bayesian model for regression under the i.i.d. data
assumption.
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David Barber. 18.1.7 by the data, we may instead approximate the above
hyperparameter integral by finding the MAP hyperparameters and use ̄f(x) ≈ ∫
f(x;w)p(w|∗,D)dw. (18.1.46) Under a flat prior p() = const., this is equivalent to
using the ...
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David Barber. 18.2 The marginal likelihood is then given by 2 logp(D|) = −β N∑ n
=1 (yn)2 + dTS−1d − log det(S) + B∑ i=1 logα i + Nlogβ − N log(2π). (18.1.52)
The EM update for β is unchanged, and the EM update for each αi is 1 αnew i + ...
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David Barber. 18.2.5 p(w|D) −1 0 1 2 −3 −2.5 −2 −1.5 −1 −0.5 0 JJ − 1 0 1 2 −3 −
2.5 −2 −1.5 −1 −0.5 0 Laplace −1 0 1 2 −3 ... Posterior approximations for a two-
dimensional Bayesian logistic regression posterior based on N = 20 datapoints.
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David Barber ... In the case of classification, no closed form Bayesian solution is
obtained by using simple Gaussian priors on the ... demoBayesLinReg.m: Demo
of Bayesian linear regression BayesLinReg.m: Bayesian linear regression ...
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David Barber. 19 Gaussian processes 19.1 19.1.1 In Bayesian linear parameter
models, we saw that the only relevant quantities are related to the scalar product
of data vectors. In Gaussian processes we use this to motivate a prediction ...
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David Barber. 19.1.2 y1yNy∗ x1xNx∗ (a) y1yNy∗ x1xNx∗ (b) Figure 19.1 (a) A
parametric model for prediction assuming i.i.d. data. (b) The form of the model
after integrating out the parameters θ. Our non-parametric model will have this ...
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David Barber. 20.7 Figure 20.22 Political books. 105 × 10 dimensional clique
matrix broken into three groups by a politically astute reader. A black square
indicates q(fic) > 0.5. Liberal books (red), conservative books (green), neutral
books ...
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David Barber. U'. (a) Mean (b) Variance ..'."'_.~.. Ii 2; I -'Z'_ [T (f) ' (E) (11) Figure
21.5 Latent identity model of face images. Each image is represented by a 70 x
70 x 3 vector (the 3 comes from the RGB colour coding). There are I = 195 ...
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David Barber. 22.1.2 δ q S Q Figure 22.1 The Rasch model for analysing
questions. Each element of the binary matrix X, with xqs = 1 if student s gets
question q correct, is generated using the latent ability of the student αs and the
latent ...
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David Barber. 23.4.4 Figure 23.13 Linear chain CRF. Since the input x is
observed, the distribution is just a linear chain factor graph. The inference of
pairwise marginals y1y2y3 p(yt, yt−1 |x) is therefore straightforward using
message passing.
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David Barber. 24.5.5 24.6 v 1 v 2 v 3 v 4 s 1 s 2 s 3 s 4 Figure 24.8 A first-order
switching AR model. In terms of inference, conditioned on v1:T, this is an HMM.
are often employed [270]. A similar situation arises in brain imaging in which
voxels ...
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David Barber. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] L. F.
Abbott, J. A. Varela, K. Sen and S. B. Nelson. Synaptic depression and cortical
gain control. Science, 275:220–223, 1997. D. H. Ackley, G. E. Hinton and T. J. ...
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D. Barber and F. V. Agakov. The IM algorithm: a variational approach to
information maximization. In Advances in Neural Information Processing Systems
(NIPS), number 16, 2004. D. Barber and C. M. Bishop. Bayesian model
comparison by ...
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David Barber. [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [54] [55] [56] [57] [59] [60
] [61] [62] [63] [64] [65] [66] C. M. Bishop. Pattern Recognition and Machine
Learning. Springer, 2006. C. M. Bishop and M. Svens ́en. Bayesian hierarchical ...
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David Barber. [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82
] [83] [84] [85] [86] [87] [88] [89] [90] [91] D. Cohn and T. Hofmann. The missing
link – a probabilistic model of document content and hypertext connectivity.
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David Barber. [92] [94] [95] [96] [97] [99] [100] [101] [102] [103] [104] [105] [106] [
107] [108] [109] [110] [111] [112] [113] [114] [115] [116] J. M. Gutierrez, E. Castillo
and A. S. Hadi. Expert Systems and Probabilistic Network Models.
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David Barber. [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153]
[154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] R.
Herbrich, T. Minka and T. Graepel. TrueSkillTM: a Bayesian skill rating system.
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David Barber. [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178]
[179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] H. J. Kappen
and W. Wiegerinck. Novel iteration schemes for the Cluster Variation Method.
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David Barber. [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202]
[203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] Y. L. Loh
, E. W. Carlson and M. Y. J. Tan. Bond-propagation algorithm for ...
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David Barber. [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227]
[228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] J. Mooij and H. J.
Kappen. Sufficient conditions for convergence of loopy belief propagation.
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Pfister, T. Toyiozumi, D. Barber and W. Gerstner. Optimal spike-timing dependent
plasticity for precise action potential firing in supervised learning. Neural
Computation, 18:1309–1339, 2006. J. Platt. Fast training of support vector
machines ...
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David Barber. [262] [263] [264] [265] [266] [267] [268] [269] [270] [271] [272] [273]
[274] [275] [276] [277] [278] [279] [280] [281] [282] [283] [284] [285] [286] M.
Seeger. Gaussian processes for machine learning. International Journal of
Neural ...
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David Barber. [287] [288] [289] [290] [291] [292] [293] [294] [295] [296] [297] [298]
[299] [300] [301] [302] [303] [304] [305] [306] [307] [308] [309] R. E. Tarjan and M.
Yannakakis. Simple linear-time algorithms to test chordality of graphs, test ...
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David Barber. [310] [311] [312] [313] [314] [315] [316] [317] [318] [319] [320] [321]
[322] [323] [324] S. Waterhouse, D. Mackay and T. Robinson. Bayesian methods
for mixtures of experts. In D. S. Touretzky, M. Mozer and M. E. Hasselmo, editors,
...
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David Barber. distribution, 173 function, 173, 202 Bethe free energy, 637 bias,
183 unbiased estimator, 183 bigram, 503 binary entropy, 624 binomial coefficient
, 171 binomial options pricing model, 150 bioinformatics, 513 black and white ...
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David Barber. empirical, 169, 310 average, 166 expectation, 166 exponential,
172 exponential family, 180 canonical, 180 ... 454 dynamic Bayesian network,
511 dynamic synapses, 580 dynamical system linear, 520 non-linear, 576
dynamics ...
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David Barber. generative approach, 315 model, 315 training, 315 generative
approach, 314 Gibbs sampling, 594 Glicko, 483 GMM, see Gaussian mixture
model Google, 491 gradient, 663 descent, 666 Gram matrix, 374 Gram–Schmidt
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David Barber. solving, 136 utility, 132 utility potential, 137 information link, 131
information maximisation, 633 information retrieval, 340, 491 information-
maximisation algorithm, 633 innovation noise, 522 input–output HMM, 509
inverse modus ...
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David Barber. probabilistic, 472 principal directions, 332 printer nightmare, 239
missing data, 281 prior, 182 ... functions, 371 Raleigh quotient, 362 Random
Boolean networks, 576 Rasch model, 479 Bayesian, 481 Rauch–Tung–Striebel,
498 ...