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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.]
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, neural nets)
Today: model-based classification with Naive Bayes
Classification
Example: Spam Filter
Input: an email
Output: spam/ham
Setup:
Get a large collection of example emails, each labeled “spam” or “ham” Note: someone has to hand label all this data! Want to learn to predict labels of new, future emails
Features: The attributes used to make the ham /
spam decision
Words: FREE! Text Patterns: $dd, CAPS Non-text: SenderInContacts, WidelyBroadcast … Dear Sir. First, I must solicit your confidence in this transaction, this is by virture of its nature as being utterly confidencial and top secret. … TO BE REMOVED FROM FUTURE MAILINGS, SIMPLY REPLY TO THIS MESSAGE AND PUT "REMOVE" IN THE SUBJECT. 99 MILLION EMAIL ADDRESSES FOR ONLY $ Ok, Iknow this is blatantly OT but I'm beginning to go insane. Had an old Dell Dimension XPS sitting in the corner and decided to put it to use, I know it was working pre being stuck in the corner, but when I plugged it in, hit the power nothing happened.
Model-Based Classification
Model-Based Classification
Model-based approach
Build a model (e.g. Bayes’ net) where
both the output label and input
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?
Naïve Bayes for Digits
Naïve Bayes: Assume all features are independent effects of the label
Simple digit recognition version:
One feature (variable) Fij 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 2 Fn
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 2 Fn |Y| parameters n x |F| x |Y| parameters |Y| x |F|n^ values
Example: Conditional Probabilities
1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 0. 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 0. 0 0.
Naïve Bayes for Text
Bag-of-words Naïve Bayes:
Features: Wi is the word at position i As before: predict label conditioned on feature variables (spam vs. ham) As before: assume features are conditionally independent given label New: each Wi 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 ith^ word in the dictionary!
Example: Spam Filtering
Model: What are the parameters?
Spam Example
- the : 0. Where do these tables come from? - to : 0. - and : 0. - of : 0. - you : 0. - a : 0. - with: 0. - from: 0. - the : 0. - to : 0. - of : 0. - 2002: 0. - with: 0. - from: 0. - and : 0. - a : 0.
- ham : 0.
- spam: 0.
- (prior) 0.33333 0.66666 - 1.1 - 0. Word P(w|spam) P(w|ham) Tot Spam Tot Ham
- Gary 0.00002 0.00021 - 11.8 - 8.
- would 0.00069 0.00084 - 19.1 - 16.
- you 0.00881 0.00304 - 23.8 - 21.
- like 0.00086 0.00083 - 30.9 - 28.
- to 0.01517 0.01339 - 35.1 - 33.
- lose 0.00008 0.00002 - 44.5 - 44.
- weight 0.00016 0.00002 - 53.3 - 55.
- while 0.00027 0.00027 - 61.5 - 63.
- you 0.00881 0.00304 - 66.2 - 69.
- sleep 0.00006 0.00001 - 76.0 - 80. - P(spam | w) = 98.
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 (many metrics possible, e.g. accuracy) 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
Generalization and Overfitting
- (^150 2 4 6 8 10 12 14 16 18 )
- 10
Degree 15 polynomial
Overfitting
Example: Overfitting
2 wins!!
Parameter Estimation
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 b r b b b^ r r (^) b b r b b
Smoothing
Maximum Likelihood?
Relative frequencies are the maximum likelihood estimates
Another option is to consider the most likely parameter value given the data
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
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 α is 0? 1?
For even better ways to estimate parameters, as well as details of
the math, see cs281a, cs
Real NB: Smoothing
For real classification problems, smoothing is critical
New odds ratios:
helvetica : 11. seems : 10. group : 10. ago : 8. areas : 8. ... verdana : 28. Credit : 28. ORDER : 27. : 26. money : 26. ...
Do these make more sense?
Tuning
Errors, and What to Do
Examples of errors
Dear GlobalSCAPE Customer, GlobalSCAPE has partnered with ScanSoft to offer you the latest version of OmniPage Pro, for just $99.99* - the regular list price is $499! The most common question we've received about this offer is - Is this genuine? We would like to assure you that this offer is authorized by ScanSoft, is genuine and valid. You can get the...
... To receive your $30 Amazon.com promotional certificate, click through to http://www.amazon.com/apparel and see the prominent link for the $30 offer. All details are there. We hope you enjoyed receiving this message. However, if you'd rather not receive future e-mails announcing new store launches, please click...
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,
and, later, how to induce new features
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
Confidences from a Classifier
The confidence of a probabilistic classifier:
Posterior probability of 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?
