Learning. Machine. A First Course in. Simon Rogers Mark Girolami. Chapman & Hall/CRC. CRC Press. Machine Learning & Pattern Recognition Series

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1 Chapman & Hall/CRC Machine Learning & Pattern Recognition Series A First Course in Machine Learning Simon Rogers Mark Girolami CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor Sc Francis Croup, an Informa business A CHAPMAN & HALL BOOK

2 a List of Tables xi List of Figures xiii Preface xix 1 Linear Modelling: A Least Squares Approach 1 11 Linear modelling Defining the model Modelling assumptions Defining what a good model is The least squares solution worked example Worked example Least squares fit to the Olympics data Summary Making predictions A second Olympics dataset Summary Vector/matrix notation Example Numerical example Making predictions Summary Nonlinear response from a linear model Generalisation and overfitting Validation data Crossvalidation Computational scaling of JiTfold crossvalidation Regularised least squares Exercises 35 Further reading 37 2 Linear Modelling: A Maximum Likelihood Approach Errors as noise Thinking generatively Random variables and probability 41 v

3 Bayes' an Olympics vi 221 Random variables Probability and distributions Adding probabilities Conditional probabilities Joint probabilities Marginalisation Aside rule 49 ' Expectations Popular discrete distributions Bernoulli distribution Binomial distribution Multinomial distribution Continuous random variables density functions Popular continuous density functions The uniform density function The beta density function The Gaussian density function Multivariate Gaussian Summary Thinking generativelycontinued Likelihood Dataset likelihood Maximum likelihood Characteristics of the maximum likelihood solution Maximum likelihood favours complex models The biasvariance tradeoff Summary 29 Effect of noise on parameter estimates Uncertainty in estimates Comparison with empirical 293 Variability in model parameters values 81 data Variability in predictions Predictive variability example Expected values of the estimators Summary Exercises 90 Further reading 93 3 The Bayesian Approach to Machine Learning A coin game Counting heads The Bayesian way The exact posterior The three scenarios No prior knowledge 104

4 the Gaussian a classconditional " vii 332 The fair coin scenario Ill 333 A biased coin The three scenarios 335 Adding more data summary Marginal likelihoods Model comparison with the marginal likelihood 35 Hyperparameters Graphical models Summary A Bayesian treatment of the Olympics 100 m data The model 38 likelihood for model 372 The likelihood The prior The posterior A firstorder polynomial Making predictions Marginal polynomial order selection 39 Chapter summary Exercises 133 Further reading Bayesian Inference Nonconjugate models Binary responses A model for binary responses A point estimate MAP solution 143 : The Laplace approximation Laplace approximation example: Approximating a gamma density Laplace approximation for the binary response model Sampling techniques 451 Playing darts The MetropolisHastings algorithm The art of sampling Summary Exercises 165 Further reading Classification The general problem Probabilistic classifiers The Bayes classifier Likelihood distributions Prior class distribution Example classconditionals 172

5 0/1 classifying the viii 5214 Making predictions The naive Bayes assumption Example text Smoothing Logistic regression Motivation Nonlinear decision functions Nonparametric models Gaussian process Nonprobabilistic classifiers ifnearest neighbours Choosing if Support vector machines and other kernel methods 5321 The margin Maximising the margin Making predictions Support vectors Soft margins Kernels Summary Assessing classification performance Accuracy 542 Sensitivity and specificity loss The area under the ROC curve Confusion matrices Discriminative and generative classifiers Summary Exercises 203 Further reading Clustering The general problem ifmeans clustering Choosing the number of clusters Where ifmeans fails Kernelised ifmeans Summary Mixture models A generative process Mixture model likelihood The EM algorithm Updating nk Updating^ Updating Sfc Updating qnk Some intuition 224

6 ix 634 Example EM finds local optima Choosing the number of components Other forms of mixture components MAP estimates with EM Bayesian mixture models Summary Exercises 234 Further reading Principal Components Analysis and Latent Variable Models The general problem Variance as a proxy for interest Principal components analysis Choosing D Limitations of PCA Latent variable models Mixture models as latent variable models Summary Variational Bayes Choosing Q{9) Optimising the bound A probabilistic model for PCA Qt(t) QXn(x ) QWm(wm) The required expectations The algorithm An example Missing values Missing values as latent variables Predicting missing values Nonrealvalued data Probit PPCA Visualising parliamentary data Aside relationship to classification Summary Exercises 273 Further reading 275 Glossary 277 Index 283

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