Pattern Recognition and Machine Learning

5.7. Bayesian Neural Networks 281 Section 3.5.3 whereγrepresents the effective number of parameters and is defined by γ= ∑W i=1 ...

282 5. NEURAL NETWORKS this framework that arise when it is applied to classification. Here we shall con- sider a network having ...

5.7. Bayesian Neural Networks 283 Figure 5.22 Illustration of the evidence framework applied to a synthetic two-class data set. ...

284 5. NEURAL NETWORKS −2 −1 0 1 2 −2 −1 0 1 2 3 −2 −1 0 1 2 −2 −1 0 1 2 3 Figure 5.23 An illustration of the Laplace approximat ...

Exercises 285 5.3 ( ) Consider a regression problem involving multiple target variables in which it is assumed that the distribu ...

286 5. NEURAL NETWORKS 5.11 ( ) www Consider a quadratic error function defined by (5.32), in which the Hessian matrixHhas an ei ...

Exercises 287 5.20 ( ) Derive an expression for the outer product approximation to the Hessian matrix for a network havingKoutpu ...

288 5. NEURAL NETWORKS components of the weight vector parallel to the eigenvectors of the Hessian satisfy w(jτ)w
j when ηj (ρτ ...

Exercises 289 5.27 ( ) www Consider the framework for training with transformed data in the special case in which the transforma ...

290 5. NEURAL NETWORKS 5.39 ( ) www Make use of the Laplace approximation result (4.135) to show that the evidence function for ...

6 Kernel Methods In Chapters 3 and 4, we considered linear parametric models for regression and classification in which the form ...

292 6. KERNEL METHODS closest example from the training set. These are examples ofmemory-basedmethods that involve storing the e ...

6.1. Dual Representations 293 6.1 Dual Representations Many linear models for regression and classification can be reformulated ...

294 6. KERNEL METHODS If we substitute this back into the linear regression model, we obtain the following prediction for a new ...

6.2. Constructing Kernels 295 −1 0 1 −1 −0.5 0 0.5 1 −1 0 1 0 0.25 0.5 0.75 1 −1 0 1 0 0.25 0.5 0.75 1 −1 0 1 0 0.02 0.04 −1 0 1 ...

296 6. KERNEL METHODS Techniques for Constructing New Kernels. Given valid kernelsk 1 (x,x′)andk 2 (x,x′), the following new ker ...

6.2. Constructing Kernels 297 omitted. We can see that this is a valid kernel by expanding the square ‖x−x′‖^2 =xTx+(x′)Tx′− 2 x ...

298 6. KERNEL METHODS This is equivalent, up to an overall multiplicative constant, to a mixture distribution in which the compo ...

6.3. Radial Basis Function Networks 299 This is the covariance matrix of the Fisher scores, and so the Fisher kernel corre- Sect ...

300 6. KERNEL METHODS point. If the differential operator is isotropic then the Green’s functions depend only on the radial dist ...