Pattern Recognition and Machine Learning
2.3. The Gaussian Distribution 101 From (2.150), we see that the effect of observingNdata points is to increase the value of the ...
102 2. PROBABILITY DISTRIBUTIONS Figure 2.14 Contour plot of the normal-gamma distribution (2.154) for parameter valuesμ 0 =0,β= ...
2.3. The Gaussian Distribution 103 Figure 2.15 Plot of Student’s t-distribution (2.159) forμ=0andλ=1for various values ofν. The ...
104 2. PROBABILITY DISTRIBUTIONS (a) −5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 (b) −5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 Figure 2.16 Illustratio ...
2.3. The Gaussian Distribution 105 St(x|μ,Λ,ν)= Γ(D/2+ν/2) Γ(ν/2) |Λ|^1 /^2 (πν)D/^2 [ 1+ ∆^2 ν ]−D/ 2 −ν/ 2 (2.162) whereDis th ...
106 2. PROBABILITY DISTRIBUTIONS Figure 2.17 Illustration of the representation of val- uesθnof a periodic variable as two- dime ...
2.3. The Gaussian Distribution 107 Figure 2.18 The von Mises distribution can be derived by considering a two-dimensional Gaussi ...
108 2. PROBABILITY DISTRIBUTIONS m=5,θ 0 =π/ 4 m=1,θ 0 =3π/ 4 2 π 0 π/ 4 3 π/ 4 m=5,θ 0 =π/ 4 m=1,θ 0 =3π/ 4 Figure 2.19 The von ...
2.3. The Gaussian Distribution 109 I 0 (m) m 0 5 10 0 1000 2000 3000 A(m) m 0 5 10 0 0.5 1 Figure 2.20 Plot of the Bessel functi ...
110 2. PROBABILITY DISTRIBUTIONS Figure 2.21 Plots of the ‘old faith- ful’ data in which the blue curves show contours of consta ...
2.3. The Gaussian Distribution 111 Figure 2.22 Example of a Gaussian mixture distribution in one dimension showing three Gaussia ...
112 2. PROBABILITY DISTRIBUTIONS 0.5 0.3 0.2 (a) 0 0.5 1 0 0.5 (^1) (b) 0 0.5 1 0 0.5 1 Figure 2.23 Illustration of a mixture of ...
2.4. The Exponential Family 113 whereX ={x 1 ,...,xN}. We immediately see that the situation is now much more complex than with ...
114 2. PROBABILITY DISTRIBUTIONS which we can solve forμto giveμ=σ(η), where σ(η)= 1 1+exp(−η) (2.199) is called thelogistic sig ...
2.4. The Exponential Family 115 Making use of the constraint (2.209), the multinomial distribution in this representa- tion then ...
116 2. PROBABILITY DISTRIBUTIONS which, after some simple rearrangement, can be cast in the standard exponential Exercise 2.57 f ...
2.4. The Exponential Family 117 which can in principle be solved to obtainηML. We see that the solution for the maximum likeliho ...
118 2. PROBABILITY DISTRIBUTIONS any subsequent observations of data. In many cases, however, we may have little idea of what fo ...
2.4. The Exponential Family 119 an intervalAμBas to the shifted intervalA−cμB−c. This implies ∫B A p(μ)dμ= ∫B−c A−c p(μ)dμ= ...
120 2. PROBABILITY DISTRIBUTIONS An example of a scale parameter would be the standard deviationσof a Gaussian distribution, aft ...
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