高斯过程机器学习概要

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"《Gaussian Processes for Machine Learning》是由Carl Edward Rasmussen和Christopher K. I. Williams合著的一本专业书籍，由麻省理工学院出版社于2006年出版，ISBN号为0-262-18253-X。这本书详细介绍了高斯过程在机器学习中的应用，涵盖了回归和分类的监督学习问题，并提供了具体的算法描述。本书是人工智能和机器学习领域的参考资料，由Adaptive Computation and Machine Learning系列编辑Thomas Dietterich策划，该系列还包括其他知名学者如Christopher Bishop、Richard S. Sutton等人的著作。" 高斯过程（Gaussian Processes, GP）是一种概率性的机器学习方法，它提供了一种基础且实用的框架，用于在基于核函数的机器学习模型中进行学习。在GP中，任何有限的输入集合都可能被看作是从一个高斯分布中随机抽样的结果。这一特性使得高斯过程在处理不确定性数据和进行非参数建模时特别有用。在回归任务中，高斯过程可以用来建立输入与输出之间的连续关系模型。通过对训练数据的学习，GP能够生成一个概率分布，表示给定新的输入时输出值的不确定性。这种概率性输出对于预测分析尤其有价值，因为它不仅给出了预测值，还提供了预测的置信度。在分类任务中，高斯过程同样可以被用来构建概率分类器。通过选择适当的核函数，GP可以捕捉特征空间中的复杂模式，从而对未标记的数据点进行分类。与传统的支持向量机（SVM）等方法相比，高斯过程分类器往往能够提供更加平滑的决策边界，并且对异常值的处理更为鲁棒。书中详细介绍了如何构建和优化高斯过程模型，包括选择合适的核函数（如径向基函数RBF）、处理大数据集时的近似方法、以及如何利用先验知识来定制模型。此外，书中还涉及了如何用马尔科夫链蒙特卡洛（MCMC）等技术进行后验推理，以求解复杂的概率模型。《Gaussian Processes for Machine Learning》一书对于理解和应用高斯过程进行机器学习是非常宝贵的资源，适合研究人员、研究生以及对机器学习有深入兴趣的从业者阅读。书中包含的实际案例和算法实现，可以帮助读者更好地理解理论概念，并将其应用于实际问题解决。同时，作者提供的在线资源www.GaussianProcess.org/gpml提供了更多的补充材料和代码，以辅助读者实践学习。

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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