2006 PatternRecognitionandMachineLea

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Subject Headings: Pattern Recognition System, Pattern Recognition Task; Pattern Recognition Algorithm; Machine Learning System; Artificial Neural Network , Bayesian Network , Sparse Kernel Machine, Textbook, Supervised Learning Algorithm, Probability Distribution Function, Linear Model Regression Algorithm, Linear Model Classification Algorithm, Neural Network Learning Algorithm, Kernel Learning Algorithm, Graphical Model Learning Algorithm, Mixture Model, EM Algorithm, Approximate Inferencing, Sampling Method, Continuous Latent Variable, Sequence Dataset Learning Algorithm, Ensemble Learning Algorithm.

Notes

Cited By

2008

Quotes

Book Overview

The dramatic growth in practical applications for machine learning over the last ten years has been accompanied by many important developments in the underlying algorithms and techniques. For example, Bayesian methods have grown from a specialist niche to become mainstream, while graphical models have emerged as a general framework for describing and applying probabilistic techniques. The practical applicability of Bayesian methods has been greatly enhanced by the development of a range of approximate inference algorithms such as variational Bayes and expectation propagation, while new models based on kernels have had a significant impact on both algorithms and applications. This completely new textbook reflects these recent developments while providing a comprehensive introduction to the fields of pattern recognition and machine learning. It is aimed at advanced undergraduates or first-year PhD students, as well as researchers and practitioners. No previous knowledge of pattern recognition or machine learning concepts is assumed. Familiarity with multivariate calculus and basic linear algebra is required, and some experience in the use of probabilities would be helpful though not essential as the book includes a self-contained introduction to basic probability theory. The book is suitable for courses on machine learning, statistics, computer science, signal processing, computer vision, data mining, and bioinformatics. Extensive support is provided for course instructors, including 431 exercises, graded according to difficulty. Example solutions for a subset of the exercises are available from the book web site, while solutions for the remainder can be obtained by instructors from the publisher. The book is supported by a great deal of additional material, and the reader is encouraged to visit the book web site for the latest information.

Abstract

This is the first textbook on pattern recognition to present the Bayesian viewpoint. The book presents approximate inference algorithms that permit fast approximate answers in situations where exact answers are not feasible. It uses graphical models to describe probability distributions when no other books apply graphical models to machine learning. No previous knowledge of pattern recognition or machine learning concepts is assumed. Familiarity with multivariate calculus and basic linear algebra is required, and some experience in the use of probabilities would be helpful though not essential as the book includes a self-contained introduction to basic probability theory.

Preface

Pattern recognition has its origins in engineering, whereas machine learning grew out of computer science. However, these activities can be viewed as two facets of the same field, and together they have undergone substantial development over the past ten years. In particular, Bayesian methods have grown from a specialist niche to become mainstream, while graphical models have emerged as a general framework for describing and applying probabilistic models. Also, the practical applicability of Bayesian methods has been greatly enhanced through the development of a range of approximate inference algorithms such as variational Bayes and expectation propagation.

Similarly, new models based on kernels have had significant impact on both algorithms and applications.

 This new textbook reflects these recent developments while providing a comprehensive introduction to the fields of pattern recognition and machine learning. It is aimed at advanced undergraduates or first year PhD students, as well as researchers and practitioners, and assumes no previous knowledge of pattern recognition or machine learning concepts. Knowledge of multivariate calculus and basic linear algebra is required, and some familiarity with probabilities would be helpful though not essential as the book includes a self-contained introduction to basic probability theory.

Because this book has broad scope, it is impossible to provide a complete list of references, and in particular no attempt has been made to provide accurate historical attribution of ideas. Instead, the aim has been to give references that offer greater detail than is possible here and that hopefully provide entry points into what, in some cases, is a very extensive literature. For this reason, the references are often to more recent textbooks and review articles rather than to original sources.

 The book is supported by a great deal of additional material, including lecture slides as well as the complete set of figures used in the book, and the reader is encouraged to visit the book web site for the latest information:

http://research.microsoft.com/∼cmbishop/PRML

Exercises

Acknowledgements

Mathematical Notation

1 Introduction

The problem of searching for patterns in data is a fundamental one and has a long and successful history. For instance, the extensive astronomical observations of Tycho Brahe in the 16th century allowed Johannes Kepler to discover the empirical laws of planetary motion, which in turn provided a springboard for the development of classical mechanics. Similarly, the discovery of regularities in atomic spectra played a key role in the development and verification of quantum physics in the early twentieth century. The field of pattern recognition is concerned with the automatic discovery of regularities in data through the use of computer algorithms and with the use of these regularities to take actions such as classifying the data into different categories.

Consider the example of recognizing handwritten digits, illustrated in Figure 1.1. Each digit corresponds to a 28 x 28 pixel image and so can be represented by a vector x comprising 784 real numbers. The goal is to build a machine that will take such a vector x as input and that will produce the identity of the digit [math]\displaystyle{ 0, … ,9 }[/math] as the output. This is a nontrivial problem due to the wide variability of handwriting. It could be tackled using handcrafted rules or heuristics for distinguishing the digits based on the shapes of the strokes, but in practice such an approach leads to a proliferation of rules and of exceptions to the rules and so on, and invariably gives poor results.

Far better results can be obtained by adopting a machine learning approach in which a large set of [math]\displaystyle{ N }[/math] digits [math]\displaystyle{ {x_1, … , X_N} }[/math] called a training set is used to tune the parameters of an adaptive model. The categories of the digits in the training set are known in advance, typically by inspecting them individually and hand-labelling them. We can express the category of a digit using target vector [math]\displaystyle{ t }[/math], which represents the identity of the corresponding digit. Suitable techniques for representing categories in terms of vectors will be discussed later. Note that there is one such target vector t for each digit image [math]\displaystyle{ x }[/math].

The result of running the machine learning algorithm can be expressed as a function [math]\displaystyle{ y(x) }[/math] which takes a new digit image x as input and that generates an output vector [math]\displaystyle{ y }[/math], encoded in the same way as the target vectors. The precise form of the function [math]\displaystyle{ y(x) }[/math] is determined during the training phase, also known as the learning phase, on the basis of the training data. Once the model is trained it can then determine the identity of new digit images, which are said to comprise a test set. The ability to categorize correctly new examples that differ from those used for training is known as generalization. In practical applications, the variability of the input vectors will be such that the training data can comprise only a tiny fraction of all possible input vectors, and so generalization is a central goal in pattern recognition.

For most practical applications, the original input variables are typically preprocessed to transform them into some new space of variables where, it is hoped, the pattern recognition problem will be easier to solve. For instance, in the digit recognition problem, the images of the digits are typically translated and scaled so that each digit is contained within a box of a fixed size. This greatly reduces the variability within each digit class, because the location and scale of all the digits are now the same, which makes it much easier for a subsequent pattern recognition algorithm to distinguish between the different classes. This pre-processing stage is sometimes also called feature extraction. Note that new test data must be pre-processed using the same steps as the training data.

 Pre-processing might also be performed in order to speed up computation. For example, if the goal is real-time face detection in a high-resolution video stream, the computer must handle huge numbers of pixels per second, and presenting these directly to a complex pattern recognition algorithm may be computationally infeasible. Instead, the aim is to find useful features that are fast to compute, and yet that also preserve useful discriminatory information enabling faces to be distinguished from non-faces. These features are then used as the inputs to the pattern recognition algorithm. For instance, the average value of the image intensity over a rectangular subregion can be evaluated extremely efficiently (Viola and Jones, 2004), and a set of such features can prove very effective in fast face detection. Because the number of such features is smaller than the number of pixels, this kind of pre-processing represents a form of dimensionality reduction. Care must be taken during pre-processing because often information is discarded, and if this information is important to the solution of the problem then the overall accuracy of the system can suffer.

Applications in which the training data comprises examples of the input vectors along with their corresponding target vectors are known as supervised learning problems. Cases such as the digit recognition example, in which the aim is to assign each input vector to one of a finite number of discrete categories, are called classification problems. If the desired output consists of one or more continuous variables, then the task is called regression. An example of a regression problem would be the prediction of the yield in a chemical manufacturing process in which the inputs consist of the concentrations of reactants, the temperature, and the pressure. In other pattern recognition problems, the training data consists of a set of input vectors x without any corresponding target values. The goal in such unsupervised learning problems may be to discover groups of similar examples within the data, where it is called clustering, or to determine the distribution of data within the input space, known as density estimation, or to project the data from a high-dimensional space down to two or three dimensions for the purpose of visualization.

Finally, the technique of reinforcement learning (Sutton and Barto, 1998) is concerned with the problem of finding suitable actions to take in a given situation in order to maximize a reward. Here the learning algorithm is not given examples of optimal outputs, in contrast to supervised learning, but must instead discover them by a process of trial and error. Typically there is a sequence of states and actions in which the learning algorithm is interacting with its environment. In many cases, the current action not only affects the immediate reward but also has an impact on there ward at all subsequent time steps. For example, by using appropriate reinforcement learning techniques a neural network can learn to play the game of backgammon to a high standard (Tesauro, 1994). Here the network must learn to take a board position as input, along with the result of a dice throw, and produce a strong move as the output. This is done by having the network play against a copy of itself for perhaps a million games. A major challenge is that a game ofbackgammon can involve dozens of moves, and yet it is only at the end of the game that the reward, in the form of victory, is achieved. The reward must then be attributed appropriately to all of the moves that led to it, even though some moves will have been good ones and others less so. This is an example of a credit assignment problem. A general feature of reinforcement learning is the trade-off between exploration, in which the system tries out new kinds of actions to see how effective they are, and exploitation, in which the system makes use of actions that are known to yield a high reward. Too strong a focus on either exploration or exploitation will yield poor results. Reinforcement learning continues to be an active area of machine learning research. However, a detailed treatment lies beyond the scope of this book.

Although each of these tasks needs its own tools and techniques, many of the key ideas that underpin them are common to all such problems. One of the main goals of this chapter is to introduce, in a relatively informal way, several of the most important of these concepts and to illustrate them using simple examples. Later in the book we shall see these same ideas re-emerge in the context of more sophisticated models that are applicable to real-world pattern recognition applications. This chapter also provides a self-contained introduction to three important tools that will be used throughout the book, namely probability theory, decision theory, and information theory. Although these might sound like daunting topics, they are in fact straightforward, and a clear understanding of them is essential if machine learning techniques are to be used to best effect in practical applications.

1.1. Example: Polynomial Curve Fitting

We begin by introducing a simple regression problem, which we shall use as a running example throughout this chapter to motivate a number of key concepts. Suppose we observe a real-valued input variable x and we wish to use this observation to predict the value of a real-valued target variable t. For the present purposes, it is instructive to consider an artificial example using synthetically generated data because we then know the precise process that generated the data for comparison against any learned model. The data for this example is generated from the function sin(21rx) with random noise included in the target values, as described in detail in Appendix A.

1.2 Probability Theory

A key concept in the field of pattern recognition is that of uncertainty. It arises both through noise among measurements, as well as through the finite size of data sets. Probability theory provides a consistent framework for the quantification and manipulation of uncertainty and forms a one of the central foundations for pattern recogtnition. When combined with decision theory, discussed in Section 1.5, it allows us to make optimal predictions given all the information available to us, even though that information may be incomplete or ambiguous.

1.3 Model Selection

1.4 The Curse of Dimensionality

1.5 Decision Theory

We have seen in Section 1.2 how probability theory provides us with a consistent mathematical framework for quantifying and manipulating uncertainty. Here we turn to the discussion of decision theory that, when combined with probability theory, allows us to make optimal decisions in situations involving uncertainty such as those encountered in pattern recognition.

Suppose we have an input vector [math]\displaystyle{ \bf{x} }[/math] together with a corresponding vector [math]\displaystyle{ \bf{t} }[/math] of target variables, and our goal is to predict [math]\displaystyle{ \bf{t} }[/math] given a new value for [math]\displaystyle{ \bf{x} }[/math]. For regression problems, [math]\displaystyle{ \bf{t} }[/math] will comprise continuous variables, whereas for classification problems [math]\displaystyle{ \bf{t} }[/math] will represent class labels. The joint probability distribution [math]\displaystyle{ p(\bf{x},\bf{t}) }[/math] provide a complete summary of the uncertainty associated with these variables. Determination of [math]\displaystyle{ p(\bf{x},\bf{t}) }[/math] from a set of training data is an example of inference and is typically a very difficult problem whose solution forms the subject of much of this book. In a practical application, however, we must often make a specific prediction of the value of [math]\displaystyle{ \bf{t} }[/math], or more generally take a specific action based on our understanding of the values [math]\displaystyle{ \bf{t} }[/math] is likely to take, and this aspect is the subject of decision theory.

2 Probability Distributions

3 Linear Models for Regression

4 Linear Models for Classification

5 Neural Networks

6 Kernel Methods

7 Sparse Kernel Machines

8 Graphical Models

Probabilities play a central role in modern pattern recognition. We have seen in Chapter 1 that probability theory can be expressed in terms of two simple equations corresponding to the sum rule and the product rule. All of the probabilistic inference and learning manipulations discussed in this book, no matter how complex, amount to repeated application of these two equations. We could therefore proceed to formulate and solve complicated probabilistic models purely by algebraic manipulation. However, we shall find it highly advantageous to augment the analysis using diagrammatic representations of probability distributions, called probabilistic graphical models. These offer several useful properties:

9 Mixture Models and EM

10 Approximate Inference

A central task in the application of probabilistic models is the evaluation of the posterior distribution p(Z|X) of the latent variables Z given the observed (visible) data variables X, and the evaluation of expectations computed with respect to this distributions. The model might also contain some determinist parameter, which we will leave implicit for the moment, or it may be a fully Bayesian model in which any unknown parameter are given prior distribution and are absorbed into the set of latent variables denoted by the vector Z. For instance, in the EM algorithm we need to evaluate the expectation of the complete-data log likelihood with respect to the posterior distribution of the latent variables. …

In such situations, we need to resort to approximation schemes, and these fall broadly into two classes, according to whether they rely on stochastic or deterministic approximations. Stochastic techniques such as Markov chain Monte Carlo, described in Chapter 11, have enables the widespread use of Bayesian methods across many domains. They generally have the property that given infinite computational resource, they can generate exact results, and the approximation arises from the use of a finite amount f processor time. In practice, sampling methods can be computationally demanding, often limited their use to small-scale problems. Also, it can be difficult to know whether a sampling scheme is generating independent samples from the require distribution.

In this chapter, we introduce a range of deterministic approximation schemes, some of which scale well to large applications. These are based on analytical approximations to the posterior distribution, for example by assuming that it factorizes in a particular way or that it has a specific parametric form such as a Gaussian. As such, they can never generate exact results, and so their strengths and weaknesses are complementary to those of sampling methods.

In Section 4.4, we discussed the Laplace approximation, which is based on a local Gaussian approximation to a mode (i.e., a maximum) of the distribution. Here we turn to a family of approximation techniques called variational inference or variational Bayes, which use more global criteria and which have been widely applied. …

10.1 Variational Inference

Variational methods have their origins in the 18th century with the work of Euler, Lagrange, and others on the “calculus of variations”. Standard calculus is concerned with finding derivatives of functions. We can think of a function as a mapping that takes the value of a variable as the input and returns the value of the function as the output. …

We can then introduce the concept of a functional derivative, which expresses how the value of the functional changes in response to infinitesimal changes to the input function. …

Although there is nothing intrinsically approximate about variational methods, they do naturally lend themselves to finding approximate solutions. This is done by restring the range of functions over which the optimization is performed, for instance by considering only quadratic functions or by considering functions composed of a linear combination of fixed basis function in which only the coefficients of the linear combinations can vary. In the case of applications to probabilistic inference, the restriction may for example take the form of factorization assumptions (Jordan et al., 1999; Jaakkola, 2001). …

10.7 Expectation Propagation

We conclude this chapter by discussion an alternative form of deterministic approximation inference, known as expectation propagation or EP (Minka, 2001a; Minka, 2001b). As with the variational Bayes methods discussed so far, this too is based on the minimization of a Kullback-Leibler divergence but now of the reverse form, which gives the approximation rather different properties.

11 Sampling Methods

12 Continuous Latent Variables

13 Sequential Data

14 Combining Models

Appendix A Data Sets

Appendix B Probability Distributions

Appendix C Properties of Matrices

Appendix D Calculus of Variations

Appendix E Lagrange Multipliers

References

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 AuthorvolumeDate ValuetitletypejournaltitleUrldoinoteyear
2006 PatternRecognitionandMachineLeaChristopher M. BishopPattern Recognition and Machine Learning2006
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