Gaussian discriminant analysis Naive Bayes

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1 DM825 Introduction to Machine Learning Lecture 7 Gaussian discriminant analysis Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark

2 Outline 1. is 2. Multi-variate Bernoulli Event Model Multinomial Event Model 3. 2

3 Discriminative approach learns p(y x) Generative approach learns p(x y) 3

4 Generative Method 1. Model p(y) and p(x y) 2. learn parameters of the models by maximizing joint likelihood p(x, y) = p(x y)p(y) 3. express 4. predict p(y x) = p(x y)p(y) p(x) = p(x y)p(y) p(x y)p(y) y Y p(x y)p(y) arg max p(y x) = arg max y y p(x) = arg max y p(x y)p(y) p(x y)p(y) y Y = arg max p(x y)p(y) y 4

5 Outline 1. is 2. Multi-variate Bernoulli Event Model Multinomial Event Model 3. 5

6 is Let x be a vector of continuous variables We will assume p( x y) is a multivariate Gaussian distribution p( x, µ, Σ) = ( ) 1 1 exp (2π) n/2 Σ 1/2 2 ( x µ)t Σ 1 ( x µ) 6

7 is Step 1: we model the probabilities: y Bernoulli(ϕ) x y = 0 N(µ 0, Σ) x y = 1 N(µ 1, Σ) that is: p(y) = φ y (1 φ) 1 y 1 p(x y = 0) = N ( µ 0, Σ) = exp ( 1 (2π) n/2 Σ 1/2 2 ( x µ 0) T Σ 1 ( x µ 0 ) ) 1 p(x y = 1) = N ( µ 1, Σ) = exp ( 1 (2π) n/2 Σ 1/2 2 ( x µ 1) T Σ 1 ( x µ 1 ) ) Step 2: we express the joint likelihood of a set of data i = 1... m: l(φ, µ 0, µ 1, Σ) = = m p(x i, y i ) i=1 m p(x i y i )p(y i ) i=1 We substitute the model assumptions above and maximize log l(φ, µ 0, µ 1, Σ) in φ, µ 0, µ 1, Σ 7

8 Solutions: m i=1 yi φ = m = i I{yi = 1} m i µ 0 = I{yi = 0}x i i I{yi = 0} i µ 1 = I{yi = 1}x i i I{yi = 1} Σ =... Compare with logistic regression where we maximized the conditional likelihood instead! Step 3 and 4: p(x y)p(y) arg max p(y x) = arg max y y p(x) arg max p(x y)p(y) y 8

9 Comments In GDA: x y Gaussian = logistic posterior for p(y = 1 x) (see sec 4.2 of B1) In logistic regression we model p(y x) as logistic also other distributions, eg: x y = 0 Poisson(λ 0 ) x y = 1 Poisson(λ 1 ) x y = 0 ExpFam(η 0 ) x y = 1 ExpFam(η 1 ) hence we make stronger assumptions in GDA. If we do not know where the data come from the logistic regression analysis would be more robust. If we know then GDA may perform better. 9

10 When Σ is the same for all class conditional densities then the decision boundaries are linear Linear discriminative analysis (LDA) When the class conditional densities do not share Σ then quadratic discriminant 10

11 Outline 1. is 2. Multi-variate Bernoulli Event Model Multinomial Event Model 3. 11

12 Chain Rule of Probability permits the calculation of the joint distribution of a set of random variables using only conditional probabilities. Consider the set of events A 1, A 2,... A n. To find the value of the joint distribution, we can apply the definition of conditional probability to obtain: Pr(A n, A n 1,..., A 1 ) = Pr(A n A n 1,..., A 1 ) Pr(A n 1, A n 2,..., A 1 ) repeating the process with each final term: n Pr( n k=1a k ) = Pr(A k k j=1a j ) k=1 For example: Pr(A 4, A 3, A 2, A 1 ) = Pr(A 4 A 3, A 2, A 1 ) Pr(A 3 A 2, A 1 ) Pr(A 2 A 1 ) Pr(A 1 ) 12

13 Multi-variate Bernoulli Event Model We want to decide whether an is spam y {0, 1} given some discrete features x. How to represent s by a set of features? Binary array, each element corresponds to a word in the vocabulary and the bit indicates whether the word is present or not in the data x = x {0, } n n = (large number) possible bit vectors parameters to learn We collect examples, look at those that are spam y = 1 and learn p(x y = 1), then at those y = 0 and learn p(x y = 0) 14

14 Step 1: For a given example i we treat each x i j independently p(y) φ y j : p(x j y = 0) φ j y=0 j : p(x j y = 1) φ j y=1 Step 2: Maximize joint likelihood Assume x j s are conditionally independent given y. By chain rule: p(x 1,..., x ) = p(x 1 y)p(x 2 y, x 1 )p(x 3 y, x 1, x 2 )... = p(x 1 y)p(x 2 y)p(x 3 y)... cond. indep. m = p(x i y) i=1 15

15 m l(φ y, φ j y=0, φ j y=1 ) = p( x i, y i ) i=1 m n = p(x i j y i )p(y i ) i=1 j=1 Solution: φ y = φ j y=1 = φ j y=0 = i I{yi = 1} m i I{yi = 1, x i j = 1} i I{yi = 1} i I{yi = 0, x i j = 1} i I{yi = 0} Step 3 and 4: prediction as usual but remember to use logarithms 16

16 Laplace Smoothing what if p(x y = 1) = 0 and p(x y = 0) = 0 because we do not have any observation in the training set with that word? p(x y = 1)p(y = 1) p(y = 1 x) = p(x y = 1)p(y = 1) + p(x y = 0)p(y = 0) j=1 p(x j y = 1)p(y = 1) = j=1 p(x j y = 1)p(y = 1) j=1 p(x j y = 0)p(y = 0) = Laplace smoothing: assume some observations p(x y) = c(x, y) + k c(y) + k x φ y = φ j y=1 = φ j y=0 = i I{yi = 1} + 1 m + K i I{yi = 1, x i j = 1} + 1 i I{yi = 1} + 2 i I{yi = 0, x i j = 1} + 1 i I{yi = 0}

17 Multinomial Event Model We look at a generalization of the previous that allows to take into account also of the number of times a word appear as well as the position. Let x i {1, 2,... K}, for example, a continuous variable discretized in buckets Let [x i 1, x i 2,... x i n i ], x i j {1, 2,..., K} represent the word in position j, n i # of word in the ith Step 1: p(y) φ y assumed that p(x j : p(x j = k y = 0) φ j = k y = 0) j y=0 is the same for all j j : p(x j = k y = 1) φ j y=1 φ j y=0 are parameters of multinomial Bernoulli distributions 19

18 Step 2: Joint likelihood: Solution: 20

19 Laplace Smoothing 21

20 Outline 1. is 2. Multi-variate Bernoulli Event Model Multinomial Event Model 3. 22

21 (Intro) Support vector machines: discriminative approach that implements a non linear decision with basis in linear classifiers. lift points to a space where they are linearly separable find linear separator Let s focus first on how to find a linear separator. Desiderata: predict 1 iff θ T x 0 predict 0 iff θ T x < 0 also wanted: if θ T x 0 very confident that y = 1 if θ T x 0 very confident that y = 0 Hence it would be nice if: i : y i = 1 we have θ T x 0 i : y i = 0 we have θ T x 0 23

22 Notation Assume training set is linearly separable Let s change notation: y { 1, 1} (instead of {0, 1} like in GLM) Let s have h output values { 1, 1}: { 1 ifz 0 f(z) = sign(z) 1 ifz < 0 (hence no probabilities like in logistic regression) h( θ, x) = f( θ x), x R n+1, θ R n+1 h( θ, x) = f( θ x + θ 0 ), x R n, θ R n, θ 0 R 24

23 Functional Marginal Def.: The functional margin of a hyperplane ( θ, θ 0 ) w.r.t. a specific example (x i, y i ) is: ˆγ i = y i ( θ T x i + θ 0 ) For the decision boundary θ T x + θ 0 that defines the linear boundary: we want θ T x 0 if y i = +1 we want θ T x 0 if y i = 1 If y i ( θ T x i + θ 0 ) > 0 then i is classified correctly. Hence, we want to maximize y i ( θ T x i + θ 0 ). This can be achieved by maximizing the worst case for the training set ˆγ = min ˆγ i i Note: scaling θ 2 θ, θ 0 θ o would make ˆγ arbitrarily large. Hence we impose: θ = 1 25

24 Hyperplanes hyperplane: set of the form { x a T x = b} ( a 0) a T x 0 x a T x = b a is the normal vector hyperplanes are affine and convex sets 26

25 Geometric Marginal Def. the geometric margin γ i is the distance of example i from the linear separator, A-B θ θ unit vector orthogonal to the separating hyperplane A-B: x i γ i θ θ since B is on the linear separator, substituting the part above in θ T x + θ 0 = 0: ( ) θ θ T x i γ i θ + θ 0 = 0 27

26 Solving for γ i : θ T x i + θ 0 = γ i θ T θ θ γ i θ = θ T x i + θ 0 θ This was for the positives. We can develop the same for the negatives but we would have a negative sign. Hence the quantity: ( θ T ) γ i = y i θ x i + θ 0 θ will be always positive. To maximize the distance of the line from all points we maximize the worst case, that is: γ = min γ i i 28

27 γ = ˆγ θ Note that if θ = 1 then ˆγ i = γ i the two marginal correspond geometric margin is invariant to scaling θ 2 θ, θ 0 θ o 29

28 Optimal margin classifier max γ θ,θ 0 γ (1) γ y i ( θ T x + θ 0 ) i = 1,..., m (2) θ = 1 (3) (2) implements γ = min γ i (3) is a nonconvex constraint thanks to which the two marginals are the same. 30