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Classification Based on Bayes Decision Theory

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Classification Based on Bayes Decision Theory Classification based on Bayes decision theory Machine Learning Hamid Beigy Sharif University of Technology Fall 1393 Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 1 / 70 1 Introduction 2 Bayes decision theory Minimizing the classification error probability Minimizing the average risk Discriminant function and decision surface Bayesian classifiers for Normally distributed classes Minimum distance classifier Bayesian classifiers for independent binary features 3 Supervised learning of the Bayesian classifiers Parametric methods for density estimation Maximum likelihood parameter estimation Bayesian estimation Maximum a posteriori estimation Mixture models for density estimation Nonparametric methods for density estimation Histogram estimator Naive estimator Kernel estimator k−Nearest neighbor estimator k−Nearest neighbor classifier 4 Naive Bayes classifier Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 2 / 70 Outline 1 Introduction 2 Bayes decision theory Minimizing the classification error probability Minimizing the average risk Discriminant function and decision surface Bayesian classifiers for Normally distributed classes Minimum distance classifier Bayesian classifiers for independent binary features 3 Supervised learning of the Bayesian classifiers Parametric methods for density estimation Maximum likelihood parameter estimation Bayesian estimation Maximum a posteriori estimation Mixture models for density estimation Nonparametric methods for density estimation Histogram estimator Naive estimator Kernel estimator k−Nearest neighbor estimator k−Nearest neighbor classifier 4 Naive Bayes classifier Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 3 / 70 Introduction In classification, the goal is to find a mapping from inputs X to outputs t given a labeled set of input-output pairs S = f(x1; t1); (x2; t2);:::; (xN ; tN )g: S is called training set. In the simplest setting, each training input x is a D−dimensional vector of numbers. Each component of x is called feature, attribute, or variable and x is called feature vector. The goal is to find a mapping from inputs X to outputs t, where t 2 f1; 2;:::; Cg with C being the number of classes. When C = 2, the problem is called binary classification. In this case, we often assume that t 2 {−1; +1g or t 2 f0; 1g. When C > 2, the problem is called multi-class classification. Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 3 / 70 Introduction (cont.) Bayes theorem P(X jC )P(C ) p(C jX ) = k k k P(X ) P(X jCk )P(Ck ) = P Y p(X jCk )p(Ck ) p(Ck ) is called prior of Ck . p(X jCk ) is called likelihood of data. p(Ck jX ) is called posterior probability. Since p(X ) is the same for all classes, we can write as p(Ck jX ) = P(X jCk )P(Ck ) Approaches for building a classifier. Generative approach: This approach first creates a joint model of the form of p(x; Ck ) and then to condition on x, then deriving p(Ck jx). Discriminative approach: This approach creates a model of the form of p(Ck jx) directly. Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 4 / 70 Outline 1 Introduction 2 Bayes decision theory Minimizing the classification error probability Minimizing the average risk Discriminant function and decision surface Bayesian classifiers for Normally distributed classes Minimum distance classifier Bayesian classifiers for independent binary features 3 Supervised learning of the Bayesian classifiers Parametric methods for density estimation Maximum likelihood parameter estimation Bayesian estimation Maximum a posteriori estimation Mixture models for density estimation Nonparametric methods for density estimation Histogram estimator Naive estimator Kernel estimator k−Nearest neighbor estimator k−Nearest neighbor classifier 4 Naive Bayes classifier Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 5 / 70 Bayes decision theory Given a classification task of M classes, C1; C2;:::; CM , and an input vector x, we can form M conditional probabilities p(Ck jx) 8k = 1; 2;:::; M Without loss of generality, consider two class classification problem. From the Bayes theorem, we have p(Ck jx) = P(xjCk )P(Ck ) The base classification rule is if p(C1jx) > p(C2jx) then x is classified to C1 if p(C1jx) < p(C2jx) then x is classified to C2 if p(C1jx) = p(C2jx) then x is classified to either C1 or C2 Since p(x) is same for all classes, then it can be removed. Hence p(xjC1)p(C1) 7 p(xjC2)p(C2) 1 If p(C1) = p(C2) = 2 , then we have p(xjC1) 7 p(xjC2) Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 5 / 70 Bayes decision theory 1 If p(C1) = p(C2) = 2 , then we have The coloured region may produce error. The probability of error equals to Pe = p(mistake) = p(x 2 R1; C2) + p(x 2 R2; C1) 1 Z 1 Z = p(xjC2)dx + p(xjC1)dx 2 R1 2 R2 Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 6 / 70 Outline 1 Introduction 2 Bayes decision theory Minimizing the classification error probability Minimizing the average risk Discriminant function and decision surface Bayesian classifiers for Normally distributed classes Minimum distance classifier Bayesian classifiers for independent binary features 3 Supervised learning of the Bayesian classifiers Parametric methods for density estimation Maximum likelihood parameter estimation Bayesian estimation Maximum a posteriori estimation Mixture models for density estimation Nonparametric methods for density estimation Histogram estimator Naive estimator Kernel estimator k−Nearest neighbor estimator k−Nearest neighbor classifier 4 Naive Bayes classifier Minimizing the classification error probability (cont.) We now show that the Bayesian classifier is optimal with respect to minimizing the classification probability. Let R1(R2) be the region in the feature space in which we decide in favor of C1 (C2). Then error is made if x 2 R1 although it belongs to C2, or if x 2 R2 but it may belongs to C1. That is Pe = p(x 2 R2; C1) + p(x 2 R1; C2) = p(x 2 R2jC1)p(C2) + p(x 2 R1jC2)p(C1) Z Z = p(C2) p(x 2 R2jC1) + p(C1) p(x 2 R1jC2) R2 R1 Since R1 [R2 covers all the feature space, from the definition of probability density function, we have Z Z p(C1) = p(C1jx)p(x)dx + p(C1jx)p(x)dx R1 R2 By combining these two equation, we obtain Z Pe = p(C1) − [p(C1jx) − p(C2jx)] p(x)dx R1 Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 7 / 70 Minimizing the classification error probability (cont.) The probability of error equals to Z Pe = p(C1) − [p(C1jx) − p(C2jx)] p(x)dx R1 The probability of error is minimized if R1 is the region of the space in which [p(C1jx) − p(C2jx)] > 0 Then R2 becomes the region where the reverse is true, i.e. is the region of the space in which [p(C1jx) − p(C2jx)] < 0 This completes the proof of the Theorem. For classification task with M classes, x is assigned to class Ck with the following rule if p(Ck jx) > p(Cj jx) 8j 6= k Show that this rule also minimizes the classification error probability for classification task with M classes. Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 8 / 70 Outline 1 Introduction 2 Bayes decision theory Minimizing the classification error probability Minimizing the average risk Discriminant function and decision surface Bayesian classifiers for Normally distributed classes Minimum distance classifier Bayesian classifiers for independent binary features 3 Supervised learning of the Bayesian classifiers Parametric methods for density estimation Maximum likelihood parameter estimation Bayesian estimation Maximum a posteriori estimation Mixture models for density estimation Nonparametric methods for density estimation Histogram estimator Naive estimator Kernel estimator k−Nearest neighbor estimator k−Nearest neighbor classifier 4 Naive Bayes classifier Minimizing the average risk The classification error probability is not always the best criterion to be adopted for minimization. Why? This because it assigns the same importance to all errors. In some applications such as IDS, patient classification, and spam filtering some wrong decisions may have more serious implications than others. In some cases, it is more appropriate to assign a penalty term to weight each error. In such case, we try to minimize the following risk. Z Z r = λ12p(C1) p(xjC1)dx + λ21 p(xjC1)dx R2 R2 In general, the risk/loss associated to class Ck is defined as M=2 X Z rk = λki p(xjCk )dx i=1 Ri The goal is to partition the feature space so that the average risk is minimized. M=2 X r = rk p(Ck ) k=1 M M ! X Z X = λki p(xjCk )p(Ck ) dx i=1 Ri k=1 Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 9 / 70 Minimizing the average risk (cont.) The average risk is equal to M M ! X Z X r = λki p(xjCk )p(Ck ) dx i=1 Ri k=1 This is achieved if each integral is minimized, so that M M X X x 2 R if ri = λki p(xjCk )p(Ck ) < rj = λkj p(xjCk )p(Ck ) 8j 6= i k=1 k=1 When λki = 1 (for k 6= i), minimizing the average risk is equivalent to minimizing the classification error probability. In two{class case, we have r1 = λ11p(xjC1)p(C1) + λ21p(xjC2)p(C2) r2 = λ12p(xjC1)p(C1) + λ22p(xjC2)p(C2) We assign x to C1 if r1 < r2, that is (λ21 − λ22) p(xjC2)p(C2) < (λ12 − λ11) p(xjC1)p(C1) Hamid Beigy (Sharif University of Technology) Classification based on Bayes decision theory Fall 1393 10 / 70 Minimizing the average risk (cont.) In other words, p(xjC1) p(C2) λ21 − λ22 x 2 C1(C2) if > (<) p(xjC2) p(C1) λ12 − λ11 Assume that the loss matrix is in the form of 0 λ Λ = 12 : λ21 0 Then, we have p(xjC1) p(C2) λ21 x 2 C1(C2) if > (<) p(xjC2) p(C1) λ12 1 When p(C1) = p(C2) = 2 , we have λ21 x 2 C1(C2) if p(xjC1) > (<)p(xjC2) λ12 If λ21 > λ12, then x is assigned to C2 if λ12 p(xjC2) > p(xjC1) λ21 That is, p(xjC1) is multiplied by a factor less than and the effect is the movement of the threshold to left of x0.
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