Pattern Recognition and Machine Learning

1.2. Probability Theory 21 1.2.3 Bayesian probabilities So far in this chapter, we have viewed probabilities in terms of the fre ...

22 1. INTRODUCTION tion of probability. Consider the example of polynomial curve fitting discussed in Section 1.1. It seems reas ...

1.2. Probability Theory 23 on this estimate are obtained by considering the distribution of possible data setsD. By contrast, fr ...

24 1. INTRODUCTION see, is required in order to make predictions or to compare different models. The development of sampling met ...

1.2. Probability Theory 25 Figure 1.13 Plot of the univariate Gaussian showing the meanμand the standard deviationσ. N(x|μ, σ^2 ...

26 1. INTRODUCTION Figure 1.14 Illustration of the likelihood function for a Gaussian distribution, shown by the red curve. Here ...

1.2. Probability Theory 27 function can be written in the form lnp ( x|μ, σ^2 ) =− 1 2 σ^2 ∑N n=1 (xn−μ)^2 − N 2 lnσ^2 − N 2 ln( ...

28 1. INTRODUCTION Figure 1.15 Illustration of how bias arises in using max- imum likelihood to determine the variance of a Gaus ...

1.2. Probability Theory 29 Figure 1.16 Schematic illustration of a Gaus- sian conditional distribution fortgivenxgiven by (1.60) ...

30 1. INTRODUCTION Again we can first determine the parameter vectorwMLgoverning the mean and sub- sequently use this to find th ...

1.2. Probability Theory 31 In the curve fitting problem, we are given the training dataxandt, along with a new test pointx, and ...

32 1. INTRODUCTION Figure 1.17 The predictive distribution result- ing from a Bayesian treatment of polynomial curve fitting usi ...

1.4. The Curse of Dimensionality 33 Figure 1.18 The technique ofS-fold cross-validation, illus- trated here for the case ofS=4, ...

34 1. INTRODUCTION Figure 1.19 Scatter plot of the oil flow data for input variablesx 6 andx 7 ,in which red denotes the ‘homoge ...

1.4. The Curse of Dimensionality 35 Figure 1.20 Illustration of a simple approach to the solution of a classification problem in ...

36 1. INTRODUCTION extend this approach to deal with input spaces having several variables. If we have Dinput variables, then a ...

1.4. The Curse of Dimensionality 37 Figure 1.22 Plot of the fraction of the volume of a sphere lying in the ranger=1− tor=1for ...

38 1. INTRODUCTION in a high-dimensional space whose dimensionality is determined by the number of pixels. Because the objects c ...

1.5. Decision Theory 39 the rest of the book. Further background, as well as more detailed accounts, can be found in Berger (198 ...

40 1. INTRODUCTION R 1 R 2 x 0 x̂ p(x,C 1 ) p(x,C 2 ) x Figure 1.24 Schematic illustration of the joint probabilitiesp(x,Ck)for ...