Generative modeling of data. Instructor: Taylor Berg-Kirkpatrick Slides: Sanjoy Dasgupta

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1 Generative modeling of data Instructor: Taylor BergKirkpatrick Slides: Sanjoy Dasgupta

2 Parametric versus nonparametric classifiers Nearest neighbor classification: size of classifier size of data set Nonparametric: Model complexity (# of parameters ) not fixed.

3 Parametric versus nonparametric classifiers Nearest neighbor classification: size of classifier size of data set Nonparametric: Model complexity (# of parameters ) not fixed. Good: Can fit any boundary, given enough data. Bad: Typically need a lot of data.

4 Parametric versus nonparametric classifiers Nearest neighbor classification: size of classifier size of data set Nonparametric: Model complexity (# of parameters ) not fixed. Good: Can fit any boundary, given enough data. Bad: Typically need a lot of data. Parametric classifiers: fixed number of parameters.

5 Parametric classifiers Fixed number of parameters: e.g. lines.

6 Parametric classifiers Fixed number of parameters: e.g. lines. Good: Can get a reasonable model with limited data Bad: Less flexibility in modeling

7 Parametric classifiers Fixed number of parameters: e.g. lines. Good: Can get a reasonable model with limited data Bad: Less flexibility in modeling Basic principle: pick a good approximation.

8 The generative approach to classification Two ways to classify: Generative: model the individual classes. Discriminative: model the decision boundary between the classes.

9 Recall: Bayes rule Pr(A B) probability event A occurs given that we have observed B to occur Pr(A B) Pr(A and B) Pr(B) Pr(B A) Pr(B) Pr(A)

10 Recall: Bayes rule Pr(A B) probability event A occurs given that we have observed B to occur Pr(A B) Pr(A and B) Pr(B) Pr(B A) Pr(B) Pr(A) Example: Coin 1 has heads probability 1/6 and Coin 2 has heads probability 1/3. We close our eyes, pick a coin at random and flip it. It comes out heads. What is the probability it is Coin 1?

11 Generative models Example: data space X R, classes/labels Y {1, 2, 3} Pr(x) P 3 (x) P 1 (x) P 2 (x) π 1 10% π 2 50% π 3 40% x

12 Generative models Example: data space X R, classes/labels Y {1, 2, 3} Pr(x) P 3 (x) P 1 (x) P 2 (x) π 1 10% π 2 50% π 3 40% x Capture the joint distribution over (x, y) via: the probabilities of the classes, π j Pr(y j) the distribution of the individual classes, P 1 (x), P 2 (x), P 3 (x)

13 Generative models Example: data space X R, classes/labels Y {1, 2, 3} Pr(x) P 3 (x) P 1 (x) P 2 (x) π 1 10% π 2 50% π 3 40% x Capture the joint distribution over (x, y) via: the probabilities of the classes, π j Pr(y j) the distribution of the individual classes, P 1 (x), P 2 (x), P 3 (x) Overall distribution: Pr(x, y) π y P y (x).

14 Optimal predictions Pr(x) P 3 (x) P 1 (x) P 2 (x) π 1 10% π 2 50% π 3 40% x

15 Optimal predictions Pr(x) P 3 (x) P 1 (x) P 2 (x) π 1 10% π 2 50% π 3 40% x For any data point x X and any candidate label j, Pr(y j x) Pr(y j)pr(x y j) Pr(x) π j P j (x) k i1 π ip i (x)

16 Optimal predictions Pr(x) P 3 (x) P 1 (x) P 2 (x) π 1 10% π 2 50% π 3 40% x For any data point x X and any candidate label j, Pr(y j x) Pr(y j)pr(x y j) Pr(x) π j P j (x) k i1 π ip i (x) Optimal prediction: the class j with largest π j P j (x).

17 A classification problem You have a bottle of wine whose label is missing. Which winery is it from, 1, 2, or 3?

18 A classification problem You have a bottle of wine whose label is missing. Which winery is it from, 1, 2, or 3? Solve this problem using visual and chemical properties of the wine.

19 The data set Training set obtained from 130 bottles Winery 1: 43 bottles Winery 2: 51 bottles Winery 3: 36 bottles For each bottle, 13 features: Alcohol, Malic acid, Ash, Alcalinity of ash, Magnesium, Total phenols, Flavanoids, Nonflavanoid phenols, Proanthocyanins, Color intensity, Hue, OD280/OD315 of diluted wines, Proline

20 The data set Training set obtained from 130 bottles Winery 1: 43 bottles Winery 2: 51 bottles Winery 3: 36 bottles For each bottle, 13 features: Alcohol, Malic acid, Ash, Alcalinity of ash, Magnesium, Total phenols, Flavanoids, Nonflavanoid phenols, Proanthocyanins, Color intensity, Hue, OD280/OD315 of diluted wines, Proline Also, a separate test set of 48 labeled points.

21 Recall: the generative approach Pr(x) P 3 (x) P 1 (x) P 2 (x) π 1 10% π 2 50% π 3 40% x For any data point x X and any candidate label j, Pr(y j x) Pr(y j)pr(x y j) Pr(x) π j P j (x) k i1 π ip i (x) Optimal prediction: the class j with largest π j P j (x).

22 Fitting a generative model Training set of 130 bottles: Winery 1: 43 bottles, winery 2: 51 bottles, winery 3: 36 bottles For each bottle, 13 features: Alcohol, Malic acid, Ash, Alcalinity of ash, Magnesium, Total phenols, Flavanoids, Nonflavanoid phenols, Proanthocyanins, Color intensity, Hue, OD280/OD315 of diluted wines, Proline

23 Fitting a generative model Training set of 130 bottles: Winery 1: 43 bottles, winery 2: 51 bottles, winery 3: 36 bottles For each bottle, 13 features: Alcohol, Malic acid, Ash, Alcalinity of ash, Magnesium, Total phenols, Flavanoids, Nonflavanoid phenols, Proanthocyanins, Color intensity, Hue, OD280/OD315 of diluted wines, Proline Class weights: π 1 43/ , π 2 51/ , π 3 36/

24 Fitting a generative model Training set of 130 bottles: Winery 1: 43 bottles, winery 2: 51 bottles, winery 3: 36 bottles For each bottle, 13 features: Alcohol, Malic acid, Ash, Alcalinity of ash, Magnesium, Total phenols, Flavanoids, Nonflavanoid phenols, Proanthocyanins, Color intensity, Hue, OD280/OD315 of diluted wines, Proline Class weights: π 1 43/ , π 2 51/ , π 3 36/ Need distributions P 1, P 2, P 3, one per class. Base these on a single feature: Alcohol.

25 The univariate Gaussian The Gaussian N(µ, σ 2 ) has mean µ, variance σ 2, and density function ) 1 (x µ)2 p(x) (2πσ 2 exp ( ) 1/2 2σ 2.

26 The distribution for winery 1 Single feature: Alcohol

27 The distribution for winery 1 Single feature: Alcohol Mean µ 13.72, Standard deviation σ 0.44 (variance 0.20)

28 All three wineries π , P 1 N(13.7, 0.20) π , P 2 N(12.3, 0.28) π , P 3 N(13.2, 0.27)

29 All three wineries π , P 1 N(13.7, 0.20) π , P 2 N(12.3, 0.28) π , P 3 N(13.2, 0.27) To classify x: Pick the j with highest π j P j (x)

30 All three wineries π , P 1 N(13.7, 0.20) π , P 2 N(12.3, 0.28) π , P 3 N(13.2, 0.27) To classify x: Pick the j with highest π j P j (x) Test error: 14/48 29%

31 All three wineries π , P 1 N(13.7, 0.20) π , P 2 N(12.3, 0.28) π , P 3 N(13.2, 0.27) To classify x: Pick the j with highest π j P j (x) Test error: 14/48 29% What if we use two features?

32 The data set, again Training set obtained from 130 bottles Winery 1: 43 bottles Winery 2: 51 bottles Winery 3: 36 bottles For each bottle, 13 features: Alcohol, Malic acid, Ash, Alcalinity of ash, Magnesium, Total phenols, Flavanoids, Nonflavanoid phenols, Proanthocyanins, Color intensity, Hue, OD280/OD315 of diluted wines, Proline Also, a separate test set of 48 labeled points.

33 The data set, again Training set obtained from 130 bottles Winery 1: 43 bottles Winery 2: 51 bottles Winery 3: 36 bottles For each bottle, 13 features: Alcohol, Malic acid, Ash, Alcalinity of ash, Magnesium, Total phenols, Flavanoids, Nonflavanoid phenols, Proanthocyanins, Color intensity, Hue, OD280/OD315 of diluted wines, Proline Also, a separate test set of 48 labeled points. This time: Alcohol and Flavanoids.

34 Why it helps to add features Better separation between the classes!

35 Why it helps to add features Better separation between the classes!

36 Why it helps to add features Better separation between the classes!

37 Why it helps to add features Better separation between the classes! Error rate drops from 29% to 8%.

38 The bivariate Gaussian

39 The bivariate Gaussian Model class 1 by a bivariate Gaussian, parametrized by: ( ) ( ) mean µ and covariance matrix Σ

40 Dependence between two random variables Suppose X 1 has mean µ 1 and X 2 has mean µ 2. Can measure dependence between them by their covariance: cov(x 1, X 2 ) E[(X 1 µ 1 )(X 2 µ 2 )] E[X 1 X 2 ] µ 1 µ 2 Maximized when X 1 X 2, in which case it is var(x 1 ). It is at most std(x 1 )std(x 2 ).

41 The bivariate (2d) Gaussian A distribution over (x 1, x 2 ) R 2, parametrized by: Mean (µ 1, µ 2 ) R 2, where µ 1 E(X 1 ) and µ 2 E(X 2 ) [ ] Σ11 Σ Covariance matrix Σ 12 where Σ 21 Σ 22 Σ 11 var(x 1 ) Σ 22 var(x 2 ) Σ 12 Σ 21 cov(x 1, X 2 )

42 The bivariate (2d) Gaussian A distribution over (x 1, x 2 ) R 2, parametrized by: Mean (µ 1, µ 2 ) R 2, where µ 1 E(X 1 ) and µ 2 E(X 2 ) [ ] Σ11 Σ Covariance matrix Σ 12 where Σ 21 Σ 22 Σ 11 var(x 1 ) Σ 22 var(x 2 ) Σ 12 Σ 21 cov(x 1, X 2 ) Density is highest at the mean, falls off in ellipsoidal contours.

43 Density of the bivariate Gaussian Mean (µ 1, µ 2 ) R 2, where µ 1 E(X 1 ) and µ 2 E(X 2 ) [ ] Σ11 Σ Covariance matrix Σ 12 Σ 21 Σ 22 ( 1 p(x 1, x 2 ) exp 1 [ ] T [ ] ) x1 µ 1 Σ 1 x1 µ 1 2π Σ 1/2 2 x 2 µ 2 x 2 µ 2

44 Bivariate Gaussian: examples In either case, the mean is (1, 1). Σ [ ] Σ [ ]

45 The decision boundary Go from 1 to 2 features: error rate goes from 29% to 8%.

46 The decision boundary Go from 1 to 2 features: error rate goes from 29% to 8%.

47 The decision boundary Go from 1 to 2 features: error rate goes from 29% to 8%. What kind of function is this? And, can we use more features?