Pattern Recognition and Machine Learning

2.3. The Gaussian Distribution 81 Figure 2.7 The red curve shows the ellip- tical surface of constant proba- bility density for ...

82 2. PROBABILITY DISTRIBUTIONS as the product of its eigenvalues, and hence |Σ|^1 /^2 = ∏D j=1 λ^1 j/^2. (2.55) Thus in theyjco ...

2.3. The Gaussian Distribution 83 where again we have changed variables usingz=x−μ. Note that the cross-terms involvingμzTandμTz ...

84 2. PROBABILITY DISTRIBUTIONS Figure 2.8 Contours of constant probability density for a Gaussian distribution in two dimension ...

2.3. The Gaussian Distribution 85 such complex distributions is that of probabilistic graphical models, which will form the subj ...

86 2. PROBABILITY DISTRIBUTIONS evaluated from the joint distributionp(x)=p(xa,xb)simply by fixingxbto the observed value and no ...

2.3. The Gaussian Distribution 87 Now consider all of the terms in (2.70) that are linear inxa xTa{Λaaμa−Λab(xb−μb)} (2.74) wher ...

88 2. PROBABILITY DISTRIBUTIONS 2.3.2 Marginal Gaussian distributions We have seen that if a joint distributionp(xa,xb)is Gaussi ...

2.3. The Gaussian Distribution 89 (2.70) that depend onxa, we obtain 1 2 [Λbbμb−Λba(xa−μa)]TΛ−bb^1 [Λbbμb−Λba(xa−μa)] − 1 2 xTaΛ ...

90 2. PROBABILITY DISTRIBUTIONS xa xb=0. 7 xb p(xa,xb) 0 0.5 1 0 0.5 1 xa p(xa) p(xa|xb=0.7) 0 0.5 1 0 5 10 Figure 2.9 The plot ...

2.3. The Gaussian Distribution 91 alinear Gaussian model(Roweis and Ghahramani, 1999), which we shall study in greater generalit ...

92 2. PROBABILITY DISTRIBUTIONS Similarly, we can find the mean of the Gaussian distribution overzby identify- ing the linear te ...

2.3. The Gaussian Distribution 93 Marginal and Conditional Gaussians Given a marginal Gaussian distribution forxand a conditiona ...

94 2. PROBABILITY DISTRIBUTIONS which is the mean of the observed set of data points. The maximization of (2.118) with respect t ...

2.3. The Gaussian Distribution 95 Figure 2.10 A schematic illustration of two correlated ran- dom variableszandθ, together with ...

96 2. PROBABILITY DISTRIBUTIONS Robbins and Monro (1951). We shall assume that the conditional variance ofzis finite so that E [ ...

2.3. The Gaussian Distribution 97 Figure 2.11 In the case of a Gaussian distribution, withθ corresponding to the meanμ, the regr ...

98 2. PROBABILITY DISTRIBUTIONS conjugate distribution for this likelihood function because the corresponding poste- rior will b ...

2.3. The Gaussian Distribution 99 Figure 2.12 Illustration of Bayesian inference for the meanμof a Gaussian distri- bution, in w ...

100 2. PROBABILITY DISTRIBUTIONS λ a=0. 1 b=0. 1 0 1 2 0 1 2 λ a=1 b=1 0 1 2 0 1 2 λ a=4 b=6 0 1 2 0 1 2 Figure 2.13 Plot of the ...