2006年MIT经典：Gaussian Processes在机器学习中的革新策略

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《高斯过程回归在机器学习中的应用》是一本由C.E. Rasmussen和C.K.I. Williams合著的经典教材，于2006年由麻省理工学院出版社（MIT Press）出版，书名全称是《Gaussian Processes for Machine Learning》。该书的国际标准书号(ISBN)为026218253X。它不仅是理论研究的重要参考资料，更是机器学习领域的实践指南，为读者提供了一种全新的机器学习方法论。高斯过程回归（Gaussian Process Regression, GPR）是本书的核心内容，它是一种非参数的概率模型，能够处理复杂且不确定性的数据，尤其在面对小样本和高维输入时，其强大的适应性和泛化能力使其在许多实际问题中表现出色。作者详细介绍了高斯过程的基本概念，包括其概率结构、推断算法以及与经典统计方法如线性回归和多项式回归的对比。此外，书中还讨论了如何通过选择合适的核函数（kernel functions）来调整模型的复杂度和表达能力。这本书不仅涵盖了理论部分，还提供了丰富的实践案例，使读者能够将高斯过程应用于诸如预测、回归分析、计算机视觉、自然语言处理等多个领域。书中包含了大量的实用代码示例和解释，帮助读者理解和掌握这一高级技术。《Gaussian Processes for Machine Learning》是适应性计算和机器学习领域的一部里程碑作品，它与其他经典的机器学习教材如《Adaptive Computation and Machine Learning》（编者Thomas Dietterich等）、《Reinforcement Learning: An Introduction》（作者Richard S. Sutton和Andrew G. Barto）等相辅相成，共同构建了机器学习的知识体系。此外，它还提到了《Bioinformatics: The Machine Learning Approach》（作者Pierre Baldi和Søren Brunak）等在生物信息学中的应用，展示了机器学习方法在不同学科中的广泛应用价值。《Gaussian Processes for Machine Learning》是一本深入浅出的教材，对于理解高斯过程的数学基础、掌握其在实际问题中的应用以及跟上当今机器学习的发展趋势具有重要意义。无论是科研人员、工程师还是学生，都能从中获益匪浅。

C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006,

ISBN 026218253X.



2006 Massachuse tts Institute of Te chnology. www.GaussianProcess.org/gpml

xvi Preface

Amos Storkey, Volker Tresp, Sethu Vijayakumar, Grace Wahba, Joe Whittaker

and Tong Zhang for valuable discussions on speciﬁc issues. We also thank Bob

Prior and the staﬀ at MIT Press for their support during the writing of the

book. We thank the Gatsby Computational Neuroscience Unit (UCL) and Neil

Lawre nce at the Department of Computer Science, University of Sheﬃeld for

hosting our visits and kindly providing space for us to work, and the Depart-

ment of Computer Science at the University of Toronto for computer support.

Thanks to John and Fiona for their hospitality on numerous occasions. Some

of the diagrams in this book have be en inspired by similar diagrams appearing

in published work, as follows: Figure 3.5, Sch¨olkopf and Smola [2002]; Fig-

ure 5.2, MacKay [1992b]. CER gratefully acknowledges ﬁnancial support from

the German Research Foundation (DFG). CKIW thanks the School of Infor-

matics, University of Edinburgh for granting him sabbatical leave for the period

October 2003-March 2004.

Finally, we reserve our deepest appreciation for our wives Agnes and Bar-

bara, and children Ezra, Kate, Miro and Ruth for their patience and under-

standing while the book was being written.

Despite our best eﬀorts it is inevitable that some errors will make it througherrata

to the printed version of the book. Errata will be made available via the book’s

we bsite at

http://www.GaussianProcess.org/gpml

We have found the joint writing of this book an excellent experience. Although

hard at times, we are conﬁdent that the end result is much better than either

one of us could have written alone.

Now, ten years after their ﬁrst introduction into the machine learning c om-looking ahead

munity, Gaussian processes are receiving growing attention. Although GPs

have been known for a long time in the statistics and geostatistics ﬁelds, and

their use can perhaps be traced back as far as the end of the 19th century, their

application to real problems is still in its early phases. This contrasts somewhat

the application of the non-probabilistic analogue of the GP, the support vec-

tor machine, which was taken up more quickly by practitioners. Perhaps this

has to do with the probabilistic mind-set needed to understand GPs, which is

not so generally appreciated. Perhaps it is due to the need for computational

short-cuts to implement inference for large datasets. Or it could be due to the

lack of a self-contained introduction to this exciting ﬁeld—with this volume, we

hope to contribute to the momentum gained by Gaussian processes in machine

learning.

Carl Edward Rasmussen and Chris Williams

T¨ubingen and Edinburgh, summer 2005

Second printing: We thank Baback Moghaddam, Mikhail Parakhin, Leif Ras-

mussen, Benjamin Sobotta, Kevin S. Van Horn and Aki Vehtari for reporting

errors in the ﬁrst printing which have now been corrected.

C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006,

ISBN 026218253X.



2006 Massachuse tts Institute of Te chnology. www.GaussianProcess.org/gpml

Symbo ls and Notation

Matrices are capitalized and vectors are in bold type. We do not generally distinguish between proba-

bilities and probability densities. A subscript asterisk, such as in X

∗

, indicates reference to a test set

quantity. A superscript asterisk denotes complex conjugate.

Symbol Meaning

\ left matrix divide: A\b is the vector x which solves Ax = b

, an equality which acts as a deﬁnition

= equality up to an additive constant

|K| determinant of K matrix

|y| Euclidean length of vector y, i.e.





1/2

hf, gi

RKHS inner pro duct

kfk

RKHS norm

the transpose of ve ctor y

∝ proportional to; e.g. p(x|y) ∝ f(x, y) means that p(x|y) is equal to f(x, y) times

a factor which is independent of x

∼ distributed according to; example: x ∼ N(µ, σ

)

∇ or ∇

partial derivatives (w.r.t. f )

∇∇ the (Hessian) matrix of second derivatives

0 or 0

vec tor of all 0’s (of length n)

1 or 1

vec tor of all 1’s (of length n)

C number of class es in a classiﬁcation problem

cholesky(A) Cholesky decomposition: L is a lower triangular matrix such that LL

= A

cov(f

∗

) Gaussian process posterior covariance

D dimension of input space X

D data set: D = {(x

, y

)|i = 1, . . . , n}

diag(w) (vec tor argument) a diagonal matrix containing the elements of vector w

diag(W ) (matrix argument) a vector containing the diagonal elements of matrix W

Kronecker delta, δ

= 1 iﬀ p = q and 0 otherwise

E or E

q( x)

[z(x)] expectation; expectation of z(x) when x ∼ q(x)

f(x) or f Gaussian process (or vector of) latent function values, f = (f (x

), . . . , f(x

))

∗

Gaussian process (posterior) prediction (random variable)

∗

Gaussian process posterior mean

GP Gaussian process: f ∼ GP



m(x), k(x, x

)



, the function f is distributed as a

Gaussian process with mean function m(x) and covariance function k(x, x

)

h(x) or h(x) either ﬁxed basis function (or set of basis functions) or weight function

H or H(X) set of basis functions evaluated at all training points

I or I

the identity matrix (of size n)

(z) Bessel function of the ﬁrst kind

k(x, x

) covariance (or kernel) function evaluated at x and x

K or K(X, X) n × n covariance (or Gram) matrix

∗

n × n

∗

matrix K(X, X

∗

), the covariance between training and test cases

k(x

∗

) or k

∗

vec tor, short for K(X, x

∗

), when there is only a single test case

or K covariance matrix for the (noise free) f values

C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006,

ISBN 026218253X.



2006 Massachuse tts Institute of Te chnology. www.GaussianProcess.org/gpml

xviii Symbols and Notation

Symbol Meaning

covariance matrix for the (noisy) y values; for independent homoscedastic noise,

= K

+ σ

(z) modiﬁed Bessel function

L(a, b) loss function, the loss of predicting b, when a is true; note argument order

log(z) natural logarithm (base e)

log

(z) logarithm to the base 2

` or `

characteristic length-scale (for input dimension d)

λ(z) logistic function, λ(z) = 1/



1 + exp(−z)



m(x) the mean function of a Gaussian process

µ a measure (see section A.7)

N(µ, Σ) or N(x|µ, Σ) (the variable x has a) Gaussian (Normal) distribution with mean vector µ and

covariance matrix Σ

N(x) short for unit Gaussian x ∼ N(0, I)

n and n

∗

number of training (and test) cases

N dimension of feature space

number of hidden units in a neural network

N the natural numbers, the positive integers

O(·) big Oh; for functions f and g on N, we write f(n) = O(g(n)) if the ratio

f(n)/g(n) remains bounded as n → ∞

O either matrix of all zeros or diﬀerential operator

y|x and p(y|x) conditional random variable y given x and its probability (density)

the regular n-polygon

φ(x

) or Φ(X) feature map of input x

(or input set X)

Φ(z) cumulative unit Gaussian: Φ(z) = (2π)

−1/2

−∞

exp(−t

/2)dt

π(x) the sigmoid of the latent value: π(x) = σ(f(x)) (stochastic if f(x) is stochastic)

ˆπ(x

∗

) MAP prediction: π evaluated at

f(x

∗

¯π(x

∗

) mean prediction: expected value of π(x

∗

). Note, in general that ˆπ(x

∗

) 6= ¯π(x

∗

)

R the real numbers

(f) or R

(l|x

∗

) expected loss for predicting l, averaged w.r.t. the model’s pred. distr. at x

∗

decision region for class c

S(s) power spectrum

σ(z) any sigmoid function, e.g. logistic λ(z), cumulative Gaussian Φ(z), etc.

variance of the (noise free) signal

noise variance

θ vec tor of hyperparameters (parameters of the c ovariance function)

tr(A) trace of (square) matrix A

the circle with circumference l

V or V

q( x)

[z(x)] variance; variance of z(x) when x ∼ q(x)

X input space and also the index set for the stochastic process

X D ×n matrix of the training inputs {x

}

i=1

: the design matrix

∗

matrix of test inputs

the ith training input

the dth coordinate of the ith training input x

Z the intege rs . . . , −2, −1, 0, 1, 2, . . .

C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006,

ISBN 026218253X.



2006 Massachuse tts Institute of Te chnology. www.GaussianProcess.org/gpml

Chapter 1

Introduction

In this book we will be concerned with supervised learning, which is the problem

of learning input-output mappings from empirical data (the training dataset).

Depending on the characteristics of the output, this problem is known as either

regression, for continuous outputs, or classiﬁcation, when outputs are discrete.

A well known example is the classiﬁcation of images of handwritten digits. digit classiﬁcation

The training se t consists of small digitized images, together with a classiﬁcation

from 0, . . . , 9, normally provided by a human. The goal is to learn a mapping

from image to classiﬁcation label, which can then be used on new, unseen

images. Supervised learning is an attractive way to attempt to tackle this

problem, since it is not easy to specify accurately the characteristics of, say, the

handwritten digit 4.

An example of a regression problem can be found in robotics, where we wish rob otic control

to learn the inverse dynamics of a robot arm. Here the task is to map from

the state of the arm (given by the positions, velocities and accelerations of the

joints) to the corresponding torques on the joints. Such a model can then be

used to compute the torques needed to move the arm along a given trajectory.

Another example would be in a chemical plant, where we might wish to predict

the yield as a function of process parameters such as temperature, pressure,

amount of catalyst e tc.

In general we denote the input as x, and the output (or target) as y. The the dataset

input is usually represented as a vector x as there are in general many input

variables—in the handwritten digit recognition example one may have a 256-

dimensional input obtained from a raster scan of a 16 × 16 image, and in the

robot arm example there are three input measurements for each joint in the

arm. The target y may either b e continuous (as in the regression case) or

discrete (as in the classiﬁcation case). We have a dataset D of n observations,

D = {(x

, y

)|i = 1, . . . , n}.

Given this training data we wish to make predictions for new inputs x

∗

training is inductive

that we have not seen in the training set. Thus it is clear that the problem

at hand is inductive; we need to move from the ﬁnite training data D to a

C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, the MIT Press, 2006,

ISBN 026218253X.



2006 Massachuse tts Institute of Te chnology. www.GaussianProcess.org/gpml

2 Introduction

function f that makes predictions for all possible input values. To do this we

must make assumptions about the characteristics of the underlying function,

as otherwise any function which is consistent with the training data would be

equally valid. A wide variety of methods have b ee n proposed to deal with the

supervised learning problem; here we describe two common approaches. Thetwo approaches

ﬁrst is to res trict the class of functions that we consider, for example by only

considering linear functions of the input. The second approach is (speaking

rather loosely) to give a prior probability to every possible function, where

higher probabilities are given to functions that we consider to be more likely, for

example because they are smoother than other functions.

The ﬁrst approach

has an obvious problem in that we have to decide upon the richness of the class

of functions considered; if we are using a mo del based on a certain class of

functions (e.g. linear functions) and the target function is not well modelled by

this class, then the predictions will be poor. One may be tempted to increase the

ﬂexibility of the class of functions, but this runs into the danger of overﬁtting,

where we can obtain a good ﬁt to the training data, but perform badly when

making test predictions.

The second approach appears to have a serious problem, in that surely

there are an uncountably inﬁnite set of possible functions, and how are we

going to compute with this set in ﬁnite time? This is where the GaussianGaussian process

process comes to our rescue. A Gaussian process is a generalization of the

Gaussian probability distribution. Whereas a probability distribution describes

random variables which are scalars or vectors (for multivariate distributions),

a stochastic process governs the properties of functions. Leaving mathematical

sophistication aside, one can loosely think of a function as a very long ve ctor,

each entry in the vector specifying the function value f(x) at a particular input

x. It turns out, that although this idea is a little na¨ıve, it is surprisingly close

what we need. Indeed, the question of how we deal computationally with these

inﬁnite dimensional objects has the most pleasant resolution imaginable: if you

ask only for the properties of the function at a ﬁnite number of points, then

inference in the Gaussian process will give you the same answer if you ignore the

inﬁnitely many other points, as if you would have taken them all into account!

And these answers are consistent with answers to any other ﬁnite queries youconsistency

may have. One of the main attractions of the Gaussian process framework is

precisely that it unites a sophisticated and consistent view with computationaltractability

tractability.

It should come as no surprise that these ideas have been around for some

time, although they are perhaps not as well known as they might be. Indeed,

many models that are commonly employed in both machine learning and statis-

tics are in fact special cases of, or restricted kinds of Gaussian processes. In this

volume, we aim to give a systematic and uniﬁed treatment of the area, showing

connections to related models.

These two app roaches may be regarded as imposing a restriction bias and a preference

bias respectively; see e.g. Mitchell [1997].

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