MLE/MAP + Naïve Bayes

Transcription

1 Introduction to Machine Learning Machine Learning Department School of Computer Science Carnegie Mellon University MLE/MAP + Naïve Bayes Matt Gormley Lecture 19 March 20,

2 Midterm Exam Reminders Thursday Evening 6:30 9:00 (2.5 hours) Room and seat assignments will be announced on Piazza You may bring one 8.5 x 11 cheatsheet 2

3 Outline Generating Data Natural (stochastic) data Synthetic data Why synthetic data? Examples: Multinomial, Bernoulli, Gaussian Data Likelihood Independent and Identically Distributed (i.i.d.) Example: Dice Rolls Learning from Data (Frequentist) Principle of Maximum Likelihood Estimation (MLE) Optimization for MLE Examples: 1D and 2D optimization Example: MLE of Multinomial Aside: Method of Lagrange Multipliers Learning from Data (Bayesian) maximum a posteriori (MAP) estimation Optimization for MAP Example: MAP of Bernoulli Beta 3

4 Whiteboard Generating Data Natural (stochastic) data Synthetic data Why synthetic data? Examples: Multinomial, Bernoulli, Gaussian 4

5 In-Class Exercise 1. With your neighbor, write a function which returns samples from a Categorical Assume access to the rand() function Function signature should be: categorical_sample(theta) where theta is the array of parameters Make your implementation as efficient as possible! 2. What is the expected runtime of your function? 5

6 Whiteboard Data Likelihood Independent and Identically Distributed (i.i.d.) Example: Dice Rolls 6

7 Learning from Data (Frequentist) Whiteboard Principle of Maximum Likelihood Estimation (MLE) Optimization for MLE Examples: 1D and 2D optimization Example: MLE of Multinomial Aside: Method of Langrange Multipliers 7

8 Learning from Data (Bayesian) Whiteboard maximum a posteriori (MAP) estimation Optimization for MAP Example: MAP of Bernoulli Beta 8

9 Takeaways One view of what ML is trying to accomplish is function approximation The principle of maximum likelihood estimation provides an alternate view of learning Synthetic data can help debug ML algorithms Probability distributions can be used to model real data that occurs in the world (don t worry we ll make our distributions more interesting soon!) 9

10 Naïve Bayes Outline Probabilistic (Generative) View of Classification Decision rule for probability model Real-world Dataset Economist vs. Onion articles Document à bag-of-words à binary feature vector Naive Bayes: Model Generating synthetic "labeled documents" Definition of model Naive Bayes assumption Counting # of parameters with / without NB assumption Naïve Bayes: Learning from Data Data likelihood MLE for Naive Bayes MAP for Naive Bayes Visualizing Gaussian Naive Bayes 10

11 Today s Goal To define a generative model of s of two different classes (e.g. spam vs. not spam) 11

12 Spam News The Economist The Onion 12

13 Whiteboard Real-world Dataset Economist vs. Onion articles Document à bag-of-words à binary feature vector 13

14 Whiteboard Naive Bayes: Model Generating synthetic "labeled documents" Definition of model Naive Bayes assumption Counting # of parameters with / without NB assumption 14

15 Model 1: Bernoulli Naïve Bayes Flip weighted coin If HEADS, flip each red coin If TAILS, flip each blue coin y x 1 x 2 x 3 x M Each red coin corresponds to an x m We can generate data in this fashion. Though in practice we never would since our data is given. Instead, this provides an explanation of how the data was generated (albeit a terrible one). 15

16 Whiteboard Naive Bayes: Model Generating synthetic "labeled documents" Definition of model Naive Bayes assumption Counting # of parameters with / without NB assumption 16

17 What s wrong with the Naïve Bayes Assumption? The features might not be independent!! Example 1: If a document contains the word Donald, it s extremely likely to contain the word Trump These are not independent! Example 2: If the petal width is very high, the petal length is also likely to be very high 17

18 Naïve Bayes: Learning from Data Whiteboard Data likelihood MLE for Naive Bayes MAP for Naive Bayes 18

19 VISUALIZING NAÏVE BAYES Slides in this section from William Cohen (10-601B, Spring 2016) 19

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21 Fisher Iris Dataset Fisher (1936) used 150 measurements of flowers from 3 different species: Iris setosa (0), Iris virginica (1), Iris versicolor (2) collected by Anderson (1936) Species Sepal Length Sepal Width Petal Length Petal Width Full dataset: 21

22 Slide from William Cohen

23 Slide from William Cohen

24 Plot the difference of the probabilities z-axis is the difference of the posterior probabilities: p(y=1 x) p(y=0 x) Slide from William Cohen

25 Question: what does the boundary between positive and negative look like for Naïve Bayes? Slide from William Cohen (10-601B, Spring 2016)

26 Iris Data (2 classes) 26

27 Iris Data (sigma not shared) 27

28 Iris Data (sigma=1) 28

29 Iris Data (3 classes) 29

30 Iris Data (sigma not shared) 30

31 Iris Data (sigma=1) 31

32 Naïve Bayes has a linear decision boundary (if sigma is shared across classes) Slide from William Cohen (10-601B, Spring 2016)

33 Figure from William Cohen (10-601B, Spring 2016)

34 Figure from William Cohen (10-601B, Spring 2016) Why don t we drop the generative model and try to learn this hyperplane directly?

35 Beyond the Scope of this Lecture Multinomial Naïve Bayes can be used for integer features Multi-class Naïve Bayes can be used if your classification problem has > 2 classes 35

36 Summary 1. Naïve Bayes provides a framework for generative modeling 2. Choose p(x m y) appropriate to the data (e.g. Bernoulli for binary features, Gaussian for continuous features) 3. Train by MLE or MAP 4. Classify by maximizing the posterior 36