UC伯克利大学的朴素贝叶斯课程

Arachchi发布于2018/06/04 00:00

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 &lt; i,j &gt; 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 ????

30.Unseen Events

31.Laplace Smoothing Laplace’s estimate: Pretend you saw every outcome once more than you actually did Can derive this estimate with Dirichlet priors (see cs281a) r r b

32.Laplace Smoothing Laplace’s estimate (extended): Pretend you saw every outcome k extra times What’s Laplace with k = 0? k is the strength of the prior Laplace for conditionals: Smooth each condition independently: r r b

33.Estimation: Linear Interpolation* In practice, Laplace often performs poorly for P(X|Y): When |X| is very large When |Y| is very large Another option: linear interpolation Also get the empirical P(X) from the data Make sure the estimate of P(X|Y) isn’t too different from the empirical P(X) What if  is 0? 1? For even better ways to estimate parameters, as well as details of the math, see cs281a, cs288

34.Real NB: Smoothing For real classification problems, smoothing is critical New odds ratios: helvetica : 11.4 seems : 10.8 group : 10.2 ago : 8.4 areas : 8.3 ... verdana : 28.8 Credit : 28.4 ORDER : 27.2 &lt;FONT&gt; : 26.9 money : 26.5 ... Do these make more sense?

35.Tuning

36.Tuning on Held-Out Data Now we’ve got two kinds of unknowns Parameters: the probabilities P(X|Y), P(Y) Hyperparameters : e.g. the amount / type of smoothing to do, k,  What should we learn where? Learn parameters from training data Tune hyperparameters on different data Why? For each value of the hyperparameters , train and test on the held-out data Choose the best value and do a final test on the test data

37.Features

39.What to Do About Errors? Need more features– words aren’t enough! Have you emailed the sender before? Have 1K other people just gotten the same email? Is the sending information consistent? Is the email in ALL CAPS? Do inline URLs point where they say they point? Does the email address you by (your) name? Can add these information sources as new variables in the NB model Next class we’ll talk about classifiers which let you easily add arbitrary features more easily

40.Baselines First step: get a baseline Baselines are very simple “straw man” procedures Help determine how hard the task is Help know what a “good” accuracy is Weak baseline: most frequent label classifier Gives all test instances whatever label was most common in the training set E.g. for spam filtering, might label everything as ham Accuracy might be very high if the problem is skewed E.g. calling everything “ham” gets 66%, so a classifier that gets 70% isn’t very good… For real research, usually use previous work as a (strong) baseline

41.Confidences from a Classifier The confidence of a probabilistic classifier: Posterior over the top label Represents how sure the classifier is of the classification Any probabilistic model will have confidences No guarantee confidence is correct Calibration Weak calibration: higher confidences mean higher accuracy Strong calibration: confidence predicts accuracy rate What’s the value of calibration?

42.Summary Bayes rule lets us do diagnostic queries with causal probabilities The naïve Bayes assumption takes all features to be independent given the class label We can build classifiers out of a naïve Bayes model using training data Smoothing estimates is important in real systems Classifier confidences are useful, when you can get them

43.Next Time: Perceptron!

44.Precision vs. Recall Let’s say we want to classify web pages as homepages or not In a test set of 1K pages, there are 3 homepages Our classifier says they are all non-homepages 99.7 accuracy! Need new measures for rare positive events Precision: fraction of guessed positives which were actually positive Recall: fraction of actual positives which were guessed as positive Say we guess 5 homepages, of which 2 were actually homepages Precision: 2 correct / 5 guessed = 0.4 Recall: 2 correct / 3 true = 0.67 Which is more important in customer support email automation? Which is more important in airport face recognition? - guessed + actual +

45.Precision vs. Recall Precision/recall tradeoff Often, you can trade off precision and recall Only works well with weakly calibrated classifiers To summarize the tradeoff: Break-even point: precision value when p = r F-measure: harmonic mean of p and r:

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