Pattern Recognition and Machine Learning

A. DATA SETS 681 oil and water, and so for infinite integration time the data will locally live on a two- dimensional manifold. ...

682 A. DATA SETS Figure A.5 Plot of the time to the next eruption in minutes (vertical axis) versus the duration of the eruption ...

A. DATA SETS 683 x t 0 1 −1 0 1 x t 0 1 −1 0 1 Figure A.6 The left-hand plot shows the synthetic regression data set along with ...

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Appendix B Probability Distributions In this appendix, we summarize the main properties of some of the most widely used probabil ...

686 B. PROBABILITY DISTRIBUTIONS Beta This is a distribution over a continuous variableμ∈[0,1], which is often used to represent ...

B. PROBABILITY DISTRIBUTIONS 687 Dirichlet The Dirichlet is a multivariate distribution overKrandom variables 0
μk
1 , wherek= ...

688 B. PROBABILITY DISTRIBUTIONS Gamma The Gamma is a probability distribution over a positive random variableτ> 0 governed b ...

B. PROBABILITY DISTRIBUTIONS 689 positive-definite. N(x|μ,Σ)= 1 (2π) D/ 2 1 |Σ|^1 /^2 exp { − 1 2 (x−μ)TΣ−^1 (x−μ) } (B.37) E[x] ...

690 B. PROBABILITY DISTRIBUTIONS and the marginal distributionp(xa)is given by p(xa)=N(xa|μa,Σaa). (B.51) Gaussian-Gamma This is ...

B. PROBABILITY DISTRIBUTIONS 691 whereIjkis thej, kelement of the identity matrix. Becausep(xk=1)=μk, the parameters must satisf ...

692 B. PROBABILITY DISTRIBUTIONS of Gaussians having the same mean but different variances. St(x|μ, λ, ν)= Γ(ν/2+1/2) Γ(ν/2) ( λ ...

B. PROBABILITY DISTRIBUTIONS 693 Von Mises The von Mises distribution, also known as the circular normal or the circular Gaus- s ...

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Appendix C Properties of Matrices In this appendix, we gather together some useful properties and identities involving matrices ...

696 C. PROPERTIES OF MATRICES which is easily proven by taking the transpose of (C.2) and applying (C.1). A useful identity invo ...

C. PROPERTIES OF MATRICES 697 to whether the permutationi 1 i 2 ...iNis even or odd, respectively. Note that|I|=1. Thus, for a 2 ...

698 C. PROPERTIES OF MATRICES Similarly ∂ ∂x (AB)= ∂A ∂x B+A ∂B ∂x . (C.20) The derivative of the inverse of a matrix can be exp ...

C. PROPERTIES OF MATRICES 699 fori=1,...,M, whereuiis aneigenvectorandλiis the correspondingeigenvalue. This can be viewed as a ...

700 C. PROPERTIES OF MATRICES and then choose the second to be orthogonal to the first (it can be shown that the de- generate ei ...