Naive Bayes介绍

UC伯克利大学的朴素贝叶斯课程
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1.Announcements Homework 9 Released soon, due Monday 4/14 at 11:59pm Final Contest (Optional) Opportunities for extra credit every Sunday Cal Day – Robot Learning Lab Open House Saturday 10am-1pm 3 rd floor Sutardja Dai Hall Robot demos of towel folding, knot tying, high-fives, fist-pumps, hugs

2.CS 188: Artificial Intelligence Naïve Bayes Instructors: Dan Klein and Pieter Abbeel --- University of California, Berkeley [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188 materials are available at http:// ai.berkeley.edu .]

3.Machine Learning Up until now: how use a model to make optimal decisions Machine learning: how to acquire a model from data / experience Learning parameters (e.g. probabilities) Learning structure (e.g. BN graphs) Learning hidden concepts (e.g. clustering) Today: model-based classification with Naive Bayes

4.Classification

5.Classification

6.Example: Digit Recognition Input: images / pixel grids Output: a digit 0-9 Setup: Get a large collection of example images, each labeled with a digit Note: someone has to hand label all this data! Want to learn to predict labels of new, future digit images Features: The attributes used to make the digit decision Pixels: (6,8)=ON Shape Patterns: NumComponents , AspectRatio , NumLoops … 0 1 2 1 ??

7.Other Classification Tasks Classification: given inputs x, predict labels (classes) y Examples: Spam detection (input: document, classes: spam / ham) OCR (input: images, classes: characters) Medical diagnosis (input: symptoms, classes: diseases) Automatic essay grading (input: document, classes: grades) Fraud detection (input: account activity, classes: fraud / no fraud) Customer service email routing … many more Classification is an important commercial technology!

8.Model-Based Classification

9.Model-Based Classification Model-based approach Build a model (e.g. Bayes’ net) where both the label and features are random variables Instantiate any observed features Query for the distribution of the label conditioned on the features Challenges What structure should the BN have? How should we learn its parameters?

10.Naïve Bayes for Digits Naïve Bayes : Assume all features are independent effects of the label Simple digit recognition version: One feature (variable) F ij for each grid position < i,j > Feature values are on / off, based on whether intensity is more or less than 0.5 in underlying image Each input maps to a feature vector, e.g. Here: lots of features, each is binary valued Naïve Bayes model: What do we need to learn? Y F 1 F n F 2

11.General Naïve Bayes A general Naive Bayes model: We only have to specify how each feature depends on the class Total number of parameters is linear in n Model is very simplistic, but often works anyway Y F 1 F n F 2 |Y| parameters n x |F| x |Y| parameters |Y| x | F| n values

12.Inference for Naïve Bayes Goal: compute posterior distribution over label variable Y Step 1: get joint probability of label and evidence for each label Step 2: sum to get probability of evidence Step 3: normalize by dividing Step 1 by Step 2 +

13.General Naïve Bayes What do we need in order to use Naïve Bayes ? Inference method (we just saw this part) Start with a bunch of probabilities: P(Y) and the P( F i |Y ) tables Use standard inference to compute P(Y|F 1 …F n ) Nothing new here Estimates of local conditional probability tables P(Y), the prior over labels P( F i |Y ) for each feature (evidence variable) These probabilities are collectively called the parameters of the model and denoted by  Up until now, we assumed these appeared by magic, but… …they typically come from training data counts: we’ll look at this soon

14.Example: Conditional Probabilities 1 0.1 2 0.1 3 0.1 4 0.1 5 0.1 6 0.1 7 0.1 8 0.1 9 0.1 0 0.1 1 0.01 2 0.05 3 0.05 4 0.30 5 0.80 6 0.90 7 0.05 8 0.60 9 0.50 0 0.80 1 0.05 2 0.01 3 0.90 4 0.80 5 0.90 6 0.90 7 0.25 8 0.85 9 0.60 0 0.80

15.Example: Conditional Probabilities 1 0.1 2 0.1 3 0.1 4 0.1 5 0.1 6 0.1 7 0.1 8 0.1 9 0.1 0 0.1 1 0.01 2 0.05 3 0.05 4 0.30 5 0.80 6 0.90 7 0.05 8 0.60 9 0.50 0 0.80 1 0.05 2 0.01 3 0.90 4 0.80 5 0.90 6 0.90 7 0.25 8 0.85 9 0.60 0 0.80

16.Naïve Bayes for Text Bag-of-words Naïve Bayes : Features: W i is the word at positon i As before: predict label conditioned on feature variables (spam vs. ham) As before: assume features are conditionally independent given label New: each W i is identically distributed Generative model: “Tied” distributions and bag-of-words Usually, each variable gets its own conditional probability distribution P(F|Y) In a bag-of-words model Each position is identically distributed All positions share the same conditional probs P(W|Y) Why make this assumption? Called “bag-of-words” because model is insensitive to word order or reordering Word at position i, not i th word in the dictionary!

17.Example: Spam Filtering Model: What are the parameters? Where do these tables come from? the : 0.0156 to : 0.0153 and : 0.0115 of : 0.0095 you : 0.0093 a : 0.0086 with: 0.0080 from: 0.0075 ... the : 0.0210 to : 0.0133 of : 0.0119 2002: 0.0110 with: 0.0108 from: 0.0107 and : 0.0105 a : 0.0100 ... ham : 0.66 spam: 0.33

18.Spam Example Word P(w|spam) P(w|ham) Tot Spam Tot Ham (prior) 0.33333 0.66666 -1.1 -0.4 Gary 0.00002 0.00021 -11.8 -8.9 would 0.00069 0.00084 -19.1 -16.0 you 0.00881 0.00304 -23.8 -21.8 like 0.00086 0.00083 -30.9 -28.9 to 0.01517 0.01339 -35.1 -33.2 lose 0.00008 0.00002 -44.5 -44.0 weight 0.00016 0.00002 -53.3 -55.0 while 0.00027 0.00027 -61.5 -63.2 you 0.00881 0.00304 -66.2 -69.0 sleep 0.00006 0.00001 -76.0 -80.5 P(spam | w) = 98.9

19.Training and Testing

20.Important Concepts Data: labeled instances, e.g. emails marked spam/ham Training set Held out set Test set Features: attribute-value pairs which characterize each x Experimentation cycle Learn parameters (e.g. model probabilities) on training set (Tune hyperparameters on held-out set) Compute accuracy of test set Very important: never “peek” at the test set! Evaluation Accuracy: fraction of instances predicted correctly Overfitting and generalization Want a classifier which does well on test data Overfitting : fitting the training data very closely, but not generalizing well We’ll investigate overfitting and generalization formally in a few lectures Training Data Held-Out Data Test Data

21.Generalization and Overfitting

22.0 2 4 6 8 10 12 14 16 18 20 -15 -10 -5 0 5 10 15 20 25 30 Degree 15 polynomial Overfitting

23.Example: Overfitting 2 wins!!

24.Example: Overfitting Posteriors determined by relative probabilities (odds ratios): south-west : inf nation : inf morally : inf nicely : inf extent : inf seriously : inf ... What went wrong here? screens : inf minute : inf guaranteed : inf $205.00 : inf delivery : inf signature : inf ...

25.Generalization and Overfitting Relative frequency parameters will overfit the training data! Just because we never saw a 3 with pixel (15,15) on during training doesn’t mean we won’t see it at test time Unlikely that every occurrence of “minute” is 100% spam Unlikely that every occurrence of “seriously” is 100% ham What about all the words that don’t occur in the training set at all? In general, we can’t go around giving unseen events zero probability As an extreme case, imagine using the entire email as the only feature Would get the training data perfect (if deterministic labeling) Wouldn’t generalize at all Just making the bag-of-words assumption gives us some generalization, but isn’t enough To generalize better: we need to smooth or regularize the estimates

26.Parameter Estimation

27.Parameter Estimation Estimating the distribution of a random variable Elicitation: ask a human (why is this hard?) Empirically: use training data (learning!) E.g.: for each outcome x, look at the empirical rate of that value: This is the estimate that maximizes the likelihood of the data r r b r b b r b b r b b r b b r b b

28.Smoothing

29.Maximum Likelihood? Relative frequencies are the maximum likelihood estimates Another option is to consider the most likely parameter value given the data ????