CSE 446: Naïve BayesWinter 2012

Dan Weld

Some slides from Carlos Guestrin, Luke Zettlemoyer & Dan Klein

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Today

• Gaussians• Naïve Bayes• Text Classification

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Long Ago• Random variables, distributions• Marginal, joint & conditional probabilities• Sum rule, product rule, Bayes rule• Independence, conditional independence

Last Time

Prior Hypothesis

Maximum Likelihood Estimate

Maximum A Posteriori Estimate

Bayesian Estimate

Uniform The most likely

Any The most likely

Any Weighted combination

Bayesian Learning

Use Bayes rule:

Or equivalently:

Prior

Normalization

Data Likelihood

Posterior

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Conjugate Priors?

Those Pesky Distributions

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Discrete Continuous

Binary {0, 1} M Values

SingleEvent

Sequence(N trials)N= H+T

Bernouilli

Binomial Multinomial

Beta DirichletConjugatePrior

What about continuous variables?

• Billionaire says: If I am measuring a continuous variable, what can you do for me?

• You say: Let me tell you about Gaussians…

Some properties of Gaussians

• Affine transformation (multiplying by scalar and adding a constant) are Gaussian– X ~ N(,2)– Y = aX + b Y ~ N(a+b,a22)

• Sum of Gaussians is Gaussian– X ~ N(X,2

X)

– Y ~ N(Y,2Y)

– Z = X+Y Z ~ N(X+Y, 2X+2

Y)

• Can make easy to differentiate, as we will see soon!

Learning a Gaussian

• Collect a bunch of data– Hopefully, i.i.d. samples– e.g., exam scores

• Learn parameters– Mean: μ– Variance: σ

Xi

i =

Exam Score

0 85

1 95

2 100

3 12

… …99 89

MLE for Gaussian:

• Prob. of i.i.d. samples D={x1,…,xN}:

• Log-likelihood of data:

Your second learning algorithm:MLE for mean of a Gaussian

• What’s MLE for mean?

MLE for variance• Again, set derivative to zero:

Learning Gaussian parameters

• MLE:

Bayesian learning of Gaussian parameters

• Conjugate priors– Mean: Gaussian prior– Variance: Wishart Distribution

Supervised Learning of ClassifiersFind f

• Given: Training set {(xi, yi) | i = 1 … n}• Find: A good approximation to f : X Y

Examples: what are X and Y ?• Spam Detection

– Map email to {Spam,Ham}

• Digit recognition– Map pixels to {0,1,2,3,4,5,6,7,8,9}

• Stock Prediction– Map new, historic prices, etc. to (the real numbers)

Classification

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Bayesian Categorization

• Let set of categories be {c1, c2,…cn}• Let E be description of an instance.• Determine category of E by determining for each ci

• P(E) can be ignored since is factor categories

)(

)|()()|(

EP

cEPcPEcP ii

i

)|()(~)|( iii cEPcPEcP

Text classification

• Classify e-mails– Y = {Spam,NotSpam}

• Classify news articles– Y = {what is the topic of the article?}

• Classify webpages– Y = {Student, professor, project, …}

• What to use for features, X?

Example: Spam Filter

• Input: 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– For email specifically,

Semantic features: SenderInContacts

– …

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 $99

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.

Features X are word sequence in document Xi for ith word in article

Features for Text Classification• X is sequence of words in document• X (and hence P(X|Y)) is huge!!!

– Article at least 1000 words, X={X1,…,X1000}

– Xi represents ith word in document, i.e., the domain of Xi is entire vocabulary, e.g., Webster Dictionary (or more), 10,000 words, etc.

• 10,0001000 = 104000

• Atoms in Universe: 1080

– We may have a problem…

Bag of Words ModelTypical additional assumption –

– Position in document doesn’t matter: • P(Xi=xi|Y=y) = P(Xk=xi|Y=y)

(all positions have the same distribution)

– Ignore the order of words

When the lecture is over, remember to wake up the person sitting next to you in the lecture room.

Bag of Words Model

in is lecture lecture next over person remember room sitting the the the to to up wake when you

Typical additional assumption –

– Position in document doesn’t matter: • P(Xi=xi|Y=y) = P(Xk=xi|Y=y)

(all position have the same distribution)

– Ignore the order of words – Sounds really silly, but often works very well!

– From now on:• Xi = Boolean: “wordi is in document”

• X = X1 … Xn

Bag of Words Approach

aardvark 0

about 2

all 2

Africa 1

apple 0

anxious 0

...

gas 1

...

oil 1

…

Zaire 0

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Bayesian Categorization

• Need to know:– Priors: P(yi)

– Conditionals: P(X | yi)

• P(yi) are easily estimated from data. – If ni of the examples in D are in yi, then P(yi) = ni / |D|

• Conditionals:– X = X1 … Xn

– Estimate P(X1 … Xn | yi)

• Too many possible instances to estimate!– (exponential in n) – Even with bag of words assumption!

Problem!

P(y1 | X) ~ P(yi)P(X|yi)

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Need to Simplify Somehow

• Too many probabilities– P(x1 x2 x3 | yi)

• Can we assume some are the same?– P(x1 x2 ) = P(x1)P(x2)

P(x1 x2 x3 | spam)P(x1 x2 x3 | spam)P(x1 x2 x3 | spam) ….P( x1 x2 x3 | spam)

?

Conditional Independence• X is conditionally independent of Y given Z, if

the probability distribution for X is independent of the value of Y, given the value of Z

• e.g.,

• Equivalent to:

Naïve Bayes

• Naïve Bayes assumption:– Features are independent given class:

– More generally:

• How many parameters now?• Suppose X is composed of n binary features

The Naïve Bayes Classifier• Given:

– Prior P(Y)

– n conditionally independent features X given the class Y

– For each Xi, we have likelihood P(Xi|

Y)

• Decision rule:

Y

X1 XnX2

MLE for the parameters of NB• Given dataset, count occurrences

• MLE for discrete NB, simply:– Prior:

– Likelihood:

Subtleties of NB Classifier 1 Violating the NB Assumption

• Usually, features are not conditionally independent:

• Actual probabilities P(Y|X) often biased towards 0 or 1• Nonetheless, NB is the single most used classifier out

there– NB often performs well, even when assumption is violated– [Domingos & Pazzani ’96] discuss some conditions for good

performance

Subtleties of NB classifier 2: Overfitting

For Binary Features: We already know the answer!

• MAP: use most likely parameter

• Beta prior equivalent to extra observations for each feature• As N → 1, prior is “forgotten”• But, for small sample size, prior is important!

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That’s Great for Bionomial

• Works for Spam / Ham• What about multiple classes

– Eg, given a wikipedia page, predicting type

Multinomials: Laplace Smoothing

• Laplace’s estimate:– Pretend you saw every outcome k

extra times

– What’s Laplace with k = 0?– k is the strength of the prior– Can derive this as a MAP estimate for

multinomial with Dirichlet priors

• Laplace for conditionals:– Smooth each condition independently:

H H T

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Naïve Bayes for Text

• Modeled as generating a bag of words for a document in a given category by repeatedly sampling with replacement from a vocabulary V = {w1, w2,…wm} based on the probabilities P(wj | ci).

• Smooth probability estimates with Laplace m-estimates assuming a uniform distribution over all words (p = 1/|V|) and m = |V|– Equivalent to a virtual sample of seeing each word in

each category exactly once.

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Easy to Implement

• But…

• If you do… it probably won’t work…

Probabilities: Important Detail!

Any more potential problems here?

• P(spam | X1 … Xn) = P(spam | Xi)i

We are multiplying lots of small numbers Danger of underflow!

0.557 = 7 E -18

Solution? Use logs and add! p1 * p2 = e log(p1)+log(p2)

Always keep in log form

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Naïve Bayes Posterior Probabilities

• Classification results of naïve Bayes – I.e. the class with maximum posterior probability…– Usually fairly accurate (?!?!?)

• However, due to the inadequacy of the conditional independence assumption…– Actual posterior-probability estimates not accurate.– Output probabilities generally very close to 0 or 1.

NB with Bag of Words for text classification

• Learning phase:– Prior P(Y)

• Count how many documents from each topic (prior)

– P(Xi|Y) • For each topic, count how many times you saw word in

documents of this topic (+ prior); remember this dist’n is shared across all positions i

• Test phase:– For each document

• Use naïve Bayes decision rule

Twenty News Groups results

Learning curve for Twenty News Groups

What if we have continuous Xi ?Eg., character recognition: Xi is ith pixel

Gaussian Naïve Bayes (GNB):

Sometimes assume variance• is independent of Y (i.e., i), • or independent of Xi (i.e., k)• or both (i.e., )

Estimating Parameters: Y discrete, Xi continuous

Maximum likelihood estimates:• Mean:

• Variance:

jth training example

(x)=1 if x true, else 0

Example: GNB for classifying mental states

~1 mm resolution

~2 images per sec.

15,000 voxels/image

non-invasive, safe

measures Blood Oxygen Level Dependent (BOLD) response

Typical impulse response

10 sec

[Mitchell et al.]

Brain scans can track activation with precision and sensitivity

[Mitchell et al.]

Gaussian Naïve Bayes: Learned voxel,word

P(BrainActivity | WordCategory = {People,Animal})[Mitchell et al.]

Learned Bayes Models – Means forP(BrainActivity | WordCategory)

Animal wordsPeople words

Pairwise classification accuracy: 85%[Mitchell et al.]

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Bayes Classifier is Optimal!• Learn: h : X Y

– X – features– Y – target classes

• Suppose: you know true P(Y|X):– Bayes classifier:

• Why?

Optimal classification

• Theorem: – Bayes classifier hBayes is optimal!

– Why?

What you need to know about Naïve Bayes

• Naïve Bayes classifier– What’s the assumption– Why we use it– How do we learn it– Why is Bayesian estimation important

• Text classification– Bag of words model

• Gaussian NB– Features are still conditionally independent– Each feature has a Gaussian distribution given class

• Optimal decision using Bayes Classifier

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Text Naïve Bayes Algorithm(Train)

Let V be the vocabulary of all words in the documents in DFor each category ci C

Let Di be the subset of documents in D in category ci

P(ci) = |Di| / |D|

Let Ti be the concatenation of all the documents in Di

Let ni be the total number of word occurrences in Ti

For each word wj V Let nij be the number of occurrences of wj in Ti

Let P(wi | ci) = (nij + 1) / (ni + |V|)

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Text Naïve Bayes Algorithm(Test)

Given a test document XLet n be the number of word occurrences in XReturn the category:

where ai is the word occurring the ith position in X

)|()(argmax1

n

iiii

CiccaPcP

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Naïve Bayes Time Complexity

• Training Time: O(|D|Ld + |C||V|)) where Ld is the average length of a document in D.– Assumes V and all Di , ni, and nij pre-computed in O(|D|

Ld) time during one pass through all of the data.– Generally just O(|D|Ld) since usually |C||V| < |D|Ld

• Test Time: O(|C| Lt) where Lt

is the average length of a test document.

• Very efficient overall, linearly proportional to the time needed to just read in all the data.

Multi-Class Categorization

• Pick the category with max probability• Create many 1 vs other classifiers• Use a hierarchical approach (wherever hierarchy

available) Entity

Person Location

Scientist Artist City County Country

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