PRML第3章PPT.pdf

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Chris Bishop’s PRML Ch. 3: Linear Models of Regression Mathieu Guillaumin & Radu Horaud October 25, 2007 Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression

Chapter content I An example – polynomial curve ﬁtting – was considered in Ch. 1 I A linear combination – regression – of a ﬁxed set of nonlinear functions – basis functions I Supervised learning: N observations {xn} with corresponding target values {tn} are provided. The goal is to predict t of a new value x. I Construct a function such that y(x) is a prediction of t. I Probabilistic perspective: model the predictive distribution p(t|x). Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression

Figure 1.16, page 29 Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression txx02σy(x0,w)y(x,w)p(t|x0,w,β)

The chapter section by section 3.1 Linear basis function models I Maximum likelihood and least squares I Geometry of least squares I Sequential learning I Regularized least squares 3.2 The bias-variance decomposition 3.3 Bayesian linear regression I Parameter distribution I Predictive distribution I Equivalent kernel 3.4 Bayesian model comparison 3.5 The evidence approximation 3.6 Limitations of ﬁxed basis functions Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression

Linear Basis Function Models M−1X j=0 y(x, w) = wjφj(x) = w>φ(x) where: I w = (w0, . . . , wM−1)> and φ = (φ0, . . . , φM−1)> with φ0(x) = 1 and w0 = bias parameter. I In general x ∈ RD but it will be convenient to treat the case x ∈ R I We observe the set X = {x1, . . . , xn, . . . , xN} with corresponding target variables t = {tn}. Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression

Basis function choices I Polynomial I Gaussian I Sigmoidal φj(x) = exp x − µj s φj(x) = σ φj(x) = xj − (x − µj)2 2s2 with σ(a) = 1 1 + e−a I splines, Fourier, wavelets, etc. Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression

Examples of basis functions Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression −101−1−0.500.51−1010 0.250.5 0.751 −10100.250.50.751

Maximum likelihood and least squares t = y(x, w) + deterministic Gaussian noise |{z} | {z } NY n=1 For a i.i.d. data set we have the likelihood function: p(t|X, w, β) = N (tn| w>φ(xn) | {z mean } , β−1|{z} var ) We can use the machinery of MLE to estimate the parameters w and the precision β: wM L = (Φ>Φ)−1Φ>t with ΦM×N = [φmn(xn)] and: tn − w> M Lφ(xn)2 NX n=1 β−1 M L = 1 N Mathieu Guillaumin & Radu Horaud Chris Bishop’s PRML Ch. 3: Linear Models of Regression

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