Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning series)

Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning series)

Carl Edward Rasmussen

Language: English

Pages: 266

ISBN: 026218253X

Format: PDF / Kindle (mobi) / ePub

Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics.The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.

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outperformed by GPR. 2.6 linear smoother Smoothing, Weight Functions and Equivalent Kernels Gaussian process regression aims to reconstruct the underlying signal f by removing the contaminating noise ε. To do this it computes a weighted average of the noisy observations y as f¯(x∗ ) = k(x∗ ) (K +σn2 I)−1 y; as f¯(x∗ ) is a linear combination of the y values, Gaussian process regression is a linear smoother (see Hastie and Tibshirani [1990, sec. 2.8] for further details). In this section we

However, as described in section 3.5.1, we can utilize the structure of W to bring down the computational load to O(Cn3 ). The Laplace approximation gives us a Gaussian approximation q(f |X, y) to the posterior p(f |X, y). To make predictions at a test point x∗ we need to compute the posterior distribution q(f∗ |X, y, x∗ ) where f (x∗ ) f∗ = (f∗1 , . . . , f∗C ) . In general we have q(f∗ |X, y, x∗ ) = p(f∗ |X, x∗ , f )q(f |X, y) df . (3.40) As p(f∗ |X, x∗ , f ) and q(f |X, y) are both

, . . .. Show that the marginal likelihood can be optimized w.r.t. θ0 in closed form. 2. Consider the difference between the log marginal likelihood given by: i log p(yi |{yj , j < i}), and the LOO-CV using log probability which is given by i log p(yi |{yj , j = i}). From the viewpoint of the marginal likelihood the LOO-CV conditions too much on the data. Show that the expected LOO-CV loss is greater than the expected marginal likelihood. Chapter 6 Relationships between GPs and Other Models In

machines that are related to Gaussian process prediction. In section 6.4 we describe support vector machines, in section 6.5 we discuss least-squares classification (LSC), and in section 6.6 we cover relevance vector machines (RVMs). 6.1 Reproducing Kernel Hilbert Spaces Here we present a brief introduction to reproducing kernel Hilbert spaces. The theory was developed by Aronszajn [1950]; a more recent treatise is Saitoh [1988]. Information can also be found in Wahba [1990], Sch¨olkopf and

then the non-separable case. We conclude this section with a comparison between GP classifiers and SVMs. ∗ 141 142 Relationships between GPs and Other Models The Separable Case Figure 6.2(a) illustrates the case where the data is linearly separable. For a linear classifier with weight vector w and offset w0 , let the decision boundary ˜ = (w, w0 ). Clearly, there is a whole be defined by w · x + w0 = 0, and let w version space of weight vectors that give rise to the same classification of

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