Dan Roth University of Illinois, Urbana-Champaign [email protected] L2R.cs.uiuc/~danr

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CS598: Machine Learning and Natural Language

Lecture 7: Probabilistic Classification

Oct. 19,21 2006

Dan RothUniversity of Illinois, [email protected]://L2R.cs.uiuc.edu/~danr

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Model the problem of text correction as that of generating correct sentences.

Goal: learn a model of the language; use it to predict.

PARADIGM Learn a probability distribution over all sentences

Use it to estimate which sentence is more likely. Pr(I saw the girl it the park) <> Pr(I saw the girl in the

park)[In the same paradigm we sometimes learn a conditional

probability distribution]

In practice: make assumptions on the distribution’s type

In practice: a decision policy depends on the assumptions

2: Generative Model

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Consider a distribution D over space XY X - the instance space; Y - set of labels. (e.g. +/-1)

Given a sample {(x,y)}1m

,, and a loss function L(x,y) Find hH that minimizes

i=1,mL(h(xi),yi) L can be: L(h(x),y)=1, h(x)y, o/w L(h(x),y) = 0 (0-1

loss)

L(h(x),y)= (h(x)-y)2 , (L2 )

L(h(x),y)=exp{- y h(x)}

Find an algorithm that minimizes average loss; then, we know that things will be okay (as a function of H).

Before: Error Driven Learning

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Goal: find the best hypothesis from some space H of hypotheses, given the observed data D.

Define best to be: most probable hypothesis in H

In order to do that, we need to assume a probability distribution over the class H.

In addition, we need to know something about the relation between the data observed and the hypotheses (E.g., a coin problem.)

As we will see, we will be Bayesian about other things, e.g., the parameters of the model

Basics of Bayesian Learning

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P(h) - the prior probability of a hypothesis h Reflects background knowledge; before data is

observed. If no information - uniform distribution.

P(D) - The probability that this sample of the Data is observed. (No knowledge of the hypothesis)

P(D|h): The probability of observing the sample D, given that the hypothesis h holds

P(h|D): The posterior probability of h. The probability h holds, given that D has been observed.

Basics of Bayesian Learning

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P(h|D) increases with P(h) and with P(D|h)

P(h|D) decreases with P(D)

P(D)P(h)h)|P(DD)|P(h

Bayes Theorem

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The learner considers a set of candidate hypotheses H (models), and attempts to find the most probable one h H, given the observed data.

Such maximally probable hypothesis is called maximum a posteriori hypothesis (MAP); Bayes theorem is used to compute it:

P(D)P(h)h)|P(DD)|P(h

h)P(h)|P(Dargmax

P(D)P(h)h)|P(DargmaxD)|P(hargmaxh

Hh

HhHhMAP

Learning Scenario

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We may assume that a priori, hypotheses are equally probable

We get the Maximum Likelihood hypothesis:

Here we just look for the hypothesis that best explains the data

h)P(h)|P(DargmaxD)|P(hargmaxh HhHhMAP

Hh,hP(hP(h jiji ),)

h)|P(Dargmaxh HhML

Learning Scenario (2)

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How should we use the general formalism? What should H be?

H can be a collection of functions. Given the training data, choose an optimal function. Then, given new data, evaluate the selected function on it.

H can be a collection of possible predictions. Given the data, try to directly choose the optimal prediction.

H can be a collection of (conditional) probability distributions.

Could be different! Specific examples we will discuss:

Naive Bayes: a maximum likelihood based algorithm; Max Entropy: seemingly, a different selection criteria; Hidden Markov Models

Bayes Optimal Classifier

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f:XV, finite set of values Instances x X can be described as a collection of features Given an example, assign it the most probable value in V

{0,1}x )x,...,x,(xx in21

),...xx,x|P(vargmax x)|P(vargmax v n21jVvjVvMAP jj

Bayesian Classifier

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• f:XV, finite set of values•Instances x X can be described as a collection of features

• Given an example, assign it the most probable value in V

• Bayes Rule:

• Notational convention: P(y) means P(Y=y)

{0,1}x )x,...,x,(xx in21

),...xx,x|P(vargmax x)|P(vargmax v n21jVvjVvMAP jj

))P(vv|,...xx,P(xargmax

),...xx,P(x

))P(vv|,...xx,P(xargmax v

jjn21Vv

n21

jjn21VvMAP

j

j

Bayesian Classifier

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• Given training data we can estimate the two terms.• Estimating P(vj) is easy. For each value vj count how many times it appears in the training data.

• However, it is not feasible to estimate

))P(vv|,...xx,P(xargmax v jjn21VvMAP j

)v|,...xx,P(x jn21

Bayesian Classifier

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• However, it is not feasible to estimate• In this case we have to estimate, for each target value, the probability of each instance (most of which will not occur)


)v|,...xx,P(x jn21

Bayesian Classifier

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• However, it is not feasible to estimate• In this case we have to estimate, for each target value, the probability of each instance (most of which will not occur)

• In order to use a Bayesian classifiers in practice, we need to make assumptions that will allow us to estimate these quantities.


)v|,...xx,P(x jn21

Bayesian Classifier

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• Assumption: feature values are independent given the target value


n

1i ji

jnjn3jn32jn21

jn3jn32jn21

jn2jn21

jn21

)v|P(x

v|P(xv|,...xP(xv,,...xx|)P(xv,,...xx|P(x

v|,...xP(xv,,...xx|)P(xv,,...xx|P(x

v|,...x)P(xv,,...xx|P(x

)v|,...xx,P(x

))...)

.......

))

)

Naive Bayes

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• Generative model:• First choose a value vj V according to P(vj )• For each vj : choose x1 x2 …, xn according to P(xk |vj )


n

1i jiijnn2211 )vv|bP(x)vv|b,...xbx,bP(x

Naive Bayes

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• Learning method: Estimate n|V| parameters and use them to compute the new value. (how to estimate?)


n


Naive Bayes

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• Learning method: Estimate n|V| parameters and use them to compute the new value.• This is learning without search. Given a collection of training examples, you just compute the best hypothesis (given the assumptions)• •This is learning without trying to achieve consistency or even approximate consistency.


n


Naive Bayes

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• Learning method: Estimate n|V| parameters and use them to compute the new value.• This is learning without search. Given a collection of training examples, you just compute the best hypothesis (given the assumptions) •This is learning without trying to achieve consistency or even approximate consistency. Why does it work?


n


Naive Bayes

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• Notice that the features values are conditionally independent, given the target value, and are not required to be independent.• Example: f(x,y)=xy over the product distribution defined by p(x=0)=p(x=1)=1/2 and p(y=0)=p(y=1)=1/2 The distribution is defined so that x and y are independent: p(x,y) = p(x)p(y) (Interpretation - for every value of x and y)• But, given that f(x,y)=0:

p(x=1|f=0) = p(y=1|f=0) = 1/3 p(x=1,y=1 | f=0) = 0

so x and y are not conditionally independent.

Conditional Independence

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• The other direction also does not hold. x and y can be conditionally independent but not independent. f=0: p(x=1|f=0) =1, p(y=1|f=0) = 0 f=1: p(x=1|f=1) =0, p(y=1|f=1) = 1 and assume, say, that p(f=0) = p(f=1)=1/2 Given the value of f, x and y are independent.• What about unconditional independence ?


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• The other direction also does not hold. x and y can be conditionally independent but not independent. f=0: p(x=1|f=0) =1, p(y=1|f=0) = 0 f=1: p(x=1|f=0) =0, p(y=1|f=1) = 1 and assume, say, that p(f=0) = p(f=1)=1/2 Given the value of f, x and y are independent.• What about unconditional independence ? p(x=1) = p(x=1|f=0)p(f=0)+p(x=1|f=1)p(f=1) = 0.5+0=0.5 p(y=1) = p(y=1|f=0)p(f=0)+p(y=1|f=1)p(f=1) = 0.5+0=0.5But, p(x=1, y=1)=p(x=1,y=1|f=0)p(f=0)+p(x=1,y=1|f=1)p(f=1) = 0

so x and y are not independent.


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Day Outlook Temperature Humidity Wind PlayTennis

1 Sunny Hot High Weak No 2 Sunny Hot High Strong No 3 Overcast Hot High Weak Yes 4 Rain Mild High Weak Yes 5 Rain Cool Normal Weak Yes 6 Rain Cool Normal Strong No 7 Overcast Cool Normal Strong Yes 8 Sunny Mild High Weak No 9 Sunny Cool Normal Weak Yes10 Rain Mild Normal Weak Yes 11 Sunny Mild Normal Strong Yes12 Overcast Mild High Strong Yes13 Overcast Hot Normal Weak Yes14 Rain Mild High Strong No

i jijVvNB )v|P(x)P(vargmax v

j

Example

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• How do we estimate P(observation | v) ?

i ino}{yes,vNB v)|nobservatioP(xP(v)argmax v

Estimating Probabilities

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• Compute P(PlayTennis= yes); P(PlayTennis= no)=• Compute P(outlook= s/oc/r | PlayTennis= yes/no) (6 numbers)• Compute P(Temp= h/mild/cool | PlayTennis= yes/no) (6 numbers)• Compute P(humidity= hi/nor | PlayTennis= yes/no) (4 numbers)• Compute P(wind= w/st | PlayTennis= yes/no) (4 numbers)


j

Example

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• Compute P(PlayTennis= yes); P(PlayTennis= no)=• Compute P(outlook= s/oc/r | PlayTennis= yes/no) (6 numbers)• Compute P(Temp= h/mild/cool | PlayTennis= yes/no) (6 numbers)• Compute P(humidity= hi/nor | PlayTennis= yes/no) (4 numbers)• Compute P(wind= w/st | PlayTennis= yes/no) (4 numbers)

•Given a new instance: (Outlook=sunny; Temperature=cool; Humidity=high; Wind=strong)

• Predict: PlayTennis= ?


j

Example

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•Given: (Outlook=sunny; Temperature=cool; Humidity=high; Wind=strong)

P(PlayTennis= yes)=9/14=0.64 P(PlayTennis= no)=5/14=0.36

P(outlook = sunny|yes)= 2/9 P(outlook = sunny|no)= 3/5 P(temp = cool | yes) = 3/9 P(temp = cool | no) = 1/5P(humidity = hi |yes) = 3/9 P(humidity = hi |no) = 4/5P(wind = strong | yes) = 3/9 P(wind = strong | no)= 3/5

P(yes|…..) ~ 0.0053 P(no|…..) ~ 0.0206


j

Example

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P(outlook = sunny|yes)= 2/9 P(outlook = sunny|no)= 3/5 P(temp = cool | yes) = 3/9 P(temp = cool | yes) = 1/5P(humidity = hi |yes) = 3/9 P(humidity = hi |yes) = 4/5P(wind = strong | yes) = 3/9 P(wind = strong | no)= 3/5

P(yes|…..) ~ 0.0053 P(no|…..) ~ 0.0206

What is we were asked about Outlook=OC ?


j

Example

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P(outlook = sunny|yes)= 2/9 P(outlook = sunny|no)= 3/5 P(temp = cool | yes) = 3/9 P(temp = cool | no) = 1/5P(humidity = hi |yes) = 3/9 P(humidity = hi |no) = 4/5P(wind = strong | yes) = 3/9 P(wind = strong | no)= 3/5

P(yes|…..) ~ 0.0053 P(no|…..) ~ 0.0206 P(no|instance) = 0/.0206/(0.0053+0.0206)=0.795


j

Example

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• Notice that the naïve Bayes method gives a method for predicting rather than an explicit classifier.• In the case of two classes, v{0,1} we predict that v=1 iff:


j

10)v|P(x0)P(v

1)v|P(x1)P(vn

1i jij

n

1i jij

Naive Bayes: Two Classes

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• Notice that the naïve Bayes method gives a method for predicting rather than an explicit classifier.• In the case of two classes, v{0,1} we predict that v=1 iff:


j

10)v|P(x0)P(v

1)v|P(x1)P(vn

1i jij

n

1i jij

1q-(1q0)P(v

p-(1p1)P(v

0)v|1p(xq 1),v|1p(xp

ii

ii

x-1i

xij

x-1i

xij

iiii

n

i

n

i

1

1

)

)

:Denote


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•In the case of two classes, v{0,1} we predict that v=1 iff:

1)

q-1

q)(q-(10)P(v

)p-1

p)(p-(11)P(v

)q-(1q0)P(v

)p-(1p1)P(v

n

1i

x

i

iij

n

1i

x

i

iij

n

1i

x-1i

xij

n

1i

x-1i

xij

i

i

ii

ii


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1)

q-1

q)(q-(10)P(v

)p-1

p)(p-(11)P(v

)q-(1q0)P(v

)p-(1p1)P(v

n

1i

x

i

iij

n

1i

x

i

iij

n

1i

x-1i

xij

n

1i

x-1i

xij

i

i

ii

ii

0)xq-1

qlog

p-1

p(log

q-1

p-1log

0)P(v

1)P(vlog

:iff 1vpredict we logarithm; Take

ii

i

ii

i

ii

i

j

j

Naïve Bayes: Two Classes

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• •We get that the optimal Bayes behavior is given by a linear separator with

1)

q-1

q)(q-(10)P(v

)p-1

p)(p-(11)P(v

)q-(1q0)P(v

)p-(1p1)P(v

n

1i

x

i

iij

n

1i

x

i

iij

n

1i

x-1i

xij

n

1i

x-1i

xij

i

i

ii

ii

0)xq-1

qlog

p-1

p(log

q-1

p-1log

0)P(v

1)P(vlog

:iff 1vpredict we logarithm; Take

ii

i

ii

i

ii

i

j

j

irrelevant is feature the and 0w then qp if

p-1

q-1

q

plog)

q-1

qlog

p-1

p(logw

iii

i

i

i

i

i

i

ii

ii


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• We have not addressed the question of why does this Classifier perform well, given that the assumptions are unlikely to be satisfied.

• The linear form of the classifiers provides some hints. (More on that later; also, Roth’99 Garg&Roth ECML’02); one of the presented papers will also address this partly.

Why does it work?

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•In the case of two classes we have that:

bxw)x |0P(v

)x |1P(vlog ii i

j

j


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•In the case of two classes we have that:

•but since

•We get (plug in (2) in (1); some algebra):

•Which is simply the logistic (sigmoid) function used in the neural network representation.

bxw)x |0P(v

)x |1P(vlog ii i

j

j

)x |0P(v-1)x |1P(v jj

b)xwexp(-1

1)x |1P(v

ii ij


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Another look at Naive Bayes

n32 x x x x1 n-1n43322 xx xx xx xx1

l)|Pr( 1

1 2 3 n

l)|Pr( n l)|Pr( 2

Pr(l)l

|} lj|j)(x, {|

|} l j 1(x) |j)(x, {|logrPlogrPlogc D

l],[D

l],[l],[ 0

ˆ/ˆ

(x)cargmax(x)prediction i

n

0il],[x0.1}{l i

|S|/|} lj|j)(x, {|logrPlogc Dl],[l],[ 00

ˆ

Note this is a bit different than the previous

linearization. Rather than a single function, here we have argmax over

several different functions.

Graphical model. It encodes the NB independence

assumption in the edge structure (siblings are

independent given parents)

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Hidden Markov Model (HMM)

HMM is a probabilistic generative model It models how an observed sequence is generated

Let’s call each position in a sequence a time step At each time step, there are two variables

Current state (hidden) Observation

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HMM

Elements Initial state probability P(s1) Transition probability P(st|st-1) Observation probability P(ot|st)

As before, the graphical model is an encoding of the independence assumptions

Note that we have seen this in the context of POS tagging.

s1

o1

s2

o2

s3

o3

s4

o4

s5

o5

s6

o6

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HMM for Shallow Parsing

States: {B, I, O}

Observations: Actual words and/or part-of-speech tags

s1=B

o1

Mr.

s2=I

o2

Brown

s3=O

o3

blamed

s4=B

o4

Mr.

s5=I

o5

Bob

s6=O

o6

for

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HMM for Shallow Parsing

Given a sentences, we can ask what the most likely state sequence is

Initial state probability:P(s1=B),P(s1=I),P(s1=O)

Transition probabilty:P(st=B|st-1=B),P(st=I|st-1=B),P(st=O|st-1=B),P(st=B|st-1=I),P(st=I|st-1=I),P(st=O|st-1=I),…

Observation Probability:P(ot=‘Mr.’|st=B),P(ot=‘Brown’|st=B),…,P(ot=‘Mr.’|st=I),P(ot=‘Brown’|st=I),…,…

s1=B

o1

Mr.

s2=I

o2

Brown

s3=O

o3

blamed

s4=B

o4

Mr.

s5=I

o5

Bob

s6=O

o6

for

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Finding most likely state sequence in HMM (1)

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A function of sk

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Viterbi’s Algorithm Dynamic Programming

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Learning the Model

Estimate Initial state probability P (s1) Transition probability P(st|st-1) Observation probability P(ot|st)

Unsupervised Learning (states are not observed) EM Algorithm

Supervised Learning (states are observed; more common)

ML Estimate of above terms directly from data

Notice that this is completely analogues to the case of naive Bayes, and essentially all other models.

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Another view of Markov Models

x)|Pr(w)t|Pr(w iii

)t,...,t,t|Pr(t)t|Pr(t 1-1ii1ii1i Assumptions:

Prediction: predict tT that maximizest)|Pr(tt)|Pr(w)t|Pr(t 1ii-1i

Input: ) ( )...t:(w?),:(w),t:),...(wt:(wx 1i1ii-1i-1i11

t t -1i t 1i

w -1i iw w 1i

States:

Observations:

T

W

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Another View of Markov Models

As for NB: features are pairs and singletons of t‘s, w’s

|} tt|)t'(t, {|

|} tt' tt |)t'(t, {|logrPlogrPlogc

1

21D]t[1,

D]t,[t]t,t[t 121121

ˆ/ˆ

(x)cargmax(x)prediction t],[xT}{t i

]t,t[t 221c t]t,[c w 0c Otherwise, t],[ ]t,t[t 121

c

Only 3 active featuresOnly 3 active features

Input: ) ( )...t:(w?),:(w),t:),...(wt:(wx 1i1ii-1i-1i11

t t -1i t 1i

w -1i iw w 1i

States:

Observations:

T

W

This can be extended to an argmax that maximizes the prediction of the whole state sequence and computed, as before, via Viterbi.

50

Learning with Probabilistic Classifiers

Learning Theory

We showed that probabilistic predictions can be viewed as predictions via Linear Statistical Queries Models (Roth’99).

The low expressivity explains Generalization+Robustness Is that all?

It does not explain why is it possible to (approximately) fit the data with these models. Namely, is there a reason to believe that these hypotheses minimize the empirical error on the sample?

In General, No. (Unless it corresponds to some probabilistic assumptions that hold).

|S| /|} lh(x)|Sx {|(h)ErrS

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Learning Protocol

LSQ hypotheses are computed directly, w/o assumptions on the underlying distribution:

- Choose features - Compute coefficients

Is there a reason to believe that an LSQ hypothesis

minimizes the empirical error on the sample?

In general, no. (Unless it corresponds to some probabilistic

assumptions that hold).

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Learning Protocol: Practice

LSQ hypotheses are computed directly: - Choose features - Compute coefficients

If hypothesis does not fit the training data - - Augment set of features

(Forget your original assumption)

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Example: probabilistic classifiers

n32 x x x x1 n-1n43322 xx xx xx xx1

Features are pairs and singletons of t‘s, w’s

Additional features are included

l)|Pr( 1

1 2 3 n

l)|Pr( n l)|Pr( 2

Pr(l)l

t t -1i t 1i

w -1i iw w 1i

States:

Observations:

T

W

If hypothesis does not fit the training data - augment set of features (forget assumptions)

54

Why is it relatively easy to fit the data? Consider all distributions with the same

marginals

(E.g, a naïve Bayes classifier will predict the same regardless of which distribution generated the data.)

(Garg&Roth ECML’01):In most cases (i.e., for most such distributions),

the resulting predictor’s error is close to optimal classifier (that if given the correct distribution)

)Pr( i l|

Robustness of Probabilistic Predictors

55

Summary: Probabilistic Modeling

Classifiers derived from probability density estimation models were viewed as LSQ

hypotheses.

Probabilistic assumptions: + Guiding feature selection but also - - Not allowing the use of more general

features.

k21 iiiiii xxxc(x)c ...

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A Unified Approach

Most methods blow up original feature space.

And make predictions using a linear representation over the new feature space

kn ) (x)... (x), (x), (x) n321 (),...,,( 321 kxxxxX

(x)ci

ijimaxarg

j

Note: Methods do not have to actually do that; But: they produce same decision as a hypothesis that does that. (Roth 98; 99,00)

57

A Unified Approach



kn ) (x)... (x), (x), (x) n321 (),...,,( 321 kxxxxX

(x)ci

ijimaxarg

j Probabilistic Methods Rule based methods

(TBL; decision lists;

exponentially decreasing weights)

Linear representation (SNoW;Perceptron;

SVM;Boosting) Memory Based Methods

(subset features)

58

A Unified Approach



kn ) (x)... (x), (x), (x) n321 (),...,,( 321 kxxxxX

(x)ci

ijimaxarg

j

Q 1: How are weights determined?Q 2: How is the new feature-space determined? Implications? Restrictions?

Documents

Dan Roth University of Illinois, Urbana-Champaign [email protected] L2R.cs.uiuc/~danr