A First Course in Machine Learning, Second Edition

"A First Course in Machine Learning by Simon Rogers and Mark Girolami is the best introductory book for ML currently available. It combines rigor and precision with accessibility, starts from a detailed explanation of the basic foundations of Bayesian analysis in the simplest of settings, and goes all the way to the frontiers of the subject such as infinite mixture models, GPs, and MCMC." —Devdatt Dubhashi, Professor, Department of Computer Science and Engineering, Chalmers University, Sweden "This textbook manages to be easier to read than other comparable books in the subject while retaining all the rigorous treatment needed. The new chapters put it at the forefront of the field by covering topics that have become mainstream in machine learning over the last decade." —Daniel Barbara, George Mason University, Fairfax, Virginia, USA "The new edition of A First Course in Machine Learning by Rogers and Girolami is an excellent introduction to the use of statistical methods in machine learning. The book introduces concepts such as mathematical modeling, inference, and prediction, providing `just in time’ the essential background on linear algebra, calculus, and probability theory that the reader needs to understand these concepts." —Daniel Ortiz-Arroyo, Associate Professor, Aalborg University Esbjerg, Denmark "I was impressed by how closely the material aligns with the needs of an introductory course on machine learning, which is its greatest strength…Overall, this is a pragmatic and helpful book, which is well-aligned to the needs of an introductory course and one that I will be looking at for my own students in coming months." —David Clifton, University of Oxford, UK "The first edition of this book was already an excellent introductory text on machine learning for an advanced undergraduate or taught masters level course, or indeed for anybody who wants to learn about an interesting and important field of computer science. The additional chapters of advanced material on Gaussian process, MCMC and mixture modeling provide an ideal basis for practical projects, without disturbing the very clear and readable exposition of the basics contained in the first part of the book." —Gavin Cawley, Senior Lecturer, School of Computing Sciences, University of East Anglia, UK "This book could be used for junior/senior undergraduate students or first-year graduate students, as well as individuals who want to explore the field of machine learning…The book introduces not only the concepts but the underlying ideas on algorithm implementation from a critical thinking perspective." —Guangzhi Qu, Oakland University, Rochester, Michigan, USA

Table of Contents:
Linear Modelling: A Least Squares Approach LINEAR MODELLING De ning the model Modelling assumptions De ning a good model The least squares solution—a worked example Worked example Least squares t to the Olympic data Summary MAKING PREDICTIONS A second Olympic dataset Summary VECTOR/MATRIX NOTATION Example Numerical example Making predictions Summary NON-LINEAR RESPONSE FROM A LINEAR MODEL GENERALISATION AND OVER-FITTING Validation data Cross-validation Computational scaling of K-fold cross-validation REGULARISED LEAST SQUARES EXERCISES FURTHER READING Linear Modelling: A Maximum Likelihood Approach ERRORS AS NOISE Thinking generatively RANDOM VARIABLES AND PROBABILITY Random variables Probability and distributions Adding probabilities Conditional probabilities Joint probabilities Marginalisation Aside—Bayes' rule 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 GENERATIVELY...CONTINUED LIKELIHOOD Dataset likelihood Maximum likelihood Characteristics of the maximum likelihood solution Maximum likelihood favours complex models THE BIAS-VARIANCE TRADE-OFF Summary EFFECT OF NOISE ON PARAMETER ESTIMATES Uncertainty in estimates Comparison with empirical values Variability in model parameters—Olympic data VARIABILITY IN PREDICTIONS Predictive variability—an example Expected values of the estimators CHAPTER SUMMARY EXERCISES FURTHER READING The Bayesian Approach to Machine Learning A COIN GAME Counting heads The Bayesian way THE EXACT POSTERIOR THE THREE SCENARIOS No prior knowledge The fair coin scenario A biased coin The three scenarios—a summary Adding more data MARGINAL LIKELIHOODS Model comparison with the marginal likelihood HYPERPARAMETERS GRAPHICAL MODELS SUMMARY A BAYESIAN TREATMENT OF THE OLYMPIC 100m DATA 122 The model The likelihood The prior The posterior A first-order polynomial Making predictions MARGINAL LIKELIHOOD FOR POLYNOMIAL MODEL ORDER SELECTION CHAPTER SUMMARY EXERCISES FURTHER READING Bayesian Inference NON-CONJUGATE MODELS BINARY RESPONSES A model for binary responses A POINT ESTIMATE—THE MAP SOLUTION THE LAPLACE APPROXIMATION Laplace approximation example: Approximating a gamma density Laplace approximation for the binary response model SAMPLING TECHNIQUES Playing darts The Metropolis{Hastings algorithm The art of sampling CHAPTER SUMMARY EXERCISES FURTHER READING Classification THE GENERAL PROBLEM PROBABILISTIC CLASSIFIERS The Bayes classifier Likelihood—class-conditional distributions Prior class distribution Example—Gaussian class-conditionals Making predictions The naive-Bayes assumption Example—classifying text Smoothing Logistic regression Motivation Non-linear decision functions Non-parametric models—the Gaussian process NON-PROBABILISTIC CLASSIFIERS K-nearest neighbours Choosing K Support vector machines and other kernel methods The margin Maximising the margin Making predictions Support vectors Soft margins Kernels Summary ASSESSING CLASSIFICATION PERFORMANCE Accuracy—0/1 loss Sensitivity and speci city The area under the ROC curve Confusion matrices DISCRIMINATIVE AND GENERATIVE CLASSIFIERS CHAPTER SUMMARY EXERCISES FURTHER READING Clustering THE GENERAL PROBLEM K-MEANS CLUSTERING Choosing the number of clusters Where K-means fails Kernelised K-means Summary MIXTURE MODELS A generative process Mixture model likelihood The EM algorithm Updating _k Updating _k Updating _k Updating qnk Some intuition Example EM nds local optima Choosing the number of components Other forms of mixture component MAP estimates with EM Bayesian mixture models CHAPTER SUMMARY EXERCISES 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(_) Optimising the bound A PROBABILISTIC MODEL FOR PCA Q_ (_ ) Qxn(xn) Qwm(wm) The required expectations The algorithm An example MISSING VALUES Missing values as latent variables Predicting missing values NON-REAL-VALUED DATA Probit PPCA Visualising parliamentary data Aside—relationship to classification CHAPTER SUMMARY EXERCISES FURTHER READING Advanced Topics Gaussian Processes PROLOGUE—NON-PARAMETRIC MODELS GAUSSIAN PROCESS REGRESSION The Gaussian process prior Noise-free regression Noisy regression Summary Noisy regression—an alternative route Alternative covariance functions Linear Polynomial Neural network ARD Composite covariance functions Summary GAUSSIAN PROCESS CLASSIFICATION A classi cation likelihood A classi cation roadmap The point estimate approximation Propagating uncertainty through the sigmoid The Laplace approximation Summary HYPERPARAMETER OPTIMISATION EXTENSIONS Non-zero mean Multiclass classi cation Other likelihood functions and models Other inference schemes CHAPTER SUMMARY EXERCISES FURTHER READING Markov Chain Monte Carlo Sampling GIBBS SAMPLING EXAMPLE: GIBBS SAMPLING FOR GP CLASSIFICATION Conditional densities for GP classi cation via Gibbs sampling Summary WHY DOES MCMC WORK? SOME SAMPLING PROBLEMS AND SOLUTIONS Burn-in and convergence Autocorrelation Summary ADVANCED SAMPLING TECHNIQUES Adaptive proposals and Hamiltonian Monte Carlo Approximate Bayesian computation Population MCMC and temperature schedules Sequential Monte Carlo CHAPTER SUMMARY EXERCISES FURTHER READING Advanced Mixture Modelling A GIBBS SAMPLER FOR MIXTURE MODELS COLLAPSED GIBBS SAMPLING AN INFINITE MIXTURE MODEL The Chinese restaurant process Inference in the in nite mixture model Summary DIRICHLET PROCESSES Hierarchical Dirichlet processes Summary BEYOND STANDARD MIXTURES—TOPIC MODELS CHAPTER SUMMARY EXERCISES FURTHER READING Glossary Index