CS60057 Speech &Natural Language Processing

Lecture 1, 7/21/2005 Natural Language Processing 1

CS60057Speech &Natural Language

Processing

Autumn 2007

Lecture 10

16 August 2007


Hidden Markov Model (HMM) Tagging

Using an HMM to do POS tagging

HMM is a special case of Bayesian inference

It is also related to the “noisy channel” model in ASR (Automatic Speech Recognition)


Goal: maximize P(word|tag) x P(tag|previous n tags)

P(word|tag) word/lexical likelihood probability that given this tag, we have this word NOT probability that this word has this tag modeled through language model (word-tag matrix)

P(tag|previous n tags) tag sequence likelihood probability that this tag follows these previous tags modeled through language model (tag-tag matrix)

Hidden Markov Model (HMM) Taggers

Lexical information Syntagmatic information


POS tagging as a sequence classification task

We are given a sentence (an “observation” or “sequence of observations”) Secretariat is expected to race tomorrow sequence of n words w1…wn.

What is the best sequence of tags which corresponds to this sequence of observations?

Probabilistic/Bayesian view: Consider all possible sequences of tags Out of this universe of sequences, choose the tag sequence

which is most probable given the observation sequence of n words w1…wn.


Getting to HMM

Let T = t1,t2,…,tn

Let W = w1,w2,…,wn

Goal: Out of all sequences of tags t1…tn, get the the most probable sequence of POS tags T underlying the observed sequence of words w1,w2,…,wn

Hat ^ means “our estimate of the best = the most probable tag sequence” Argmaxx f(x) means “the x such that f(x) is maximized”

it maximazes our estimate of the best tag sequence


Getting to HMM

This equation is guaranteed to give us the best tag sequence

But how do we make it operational? How do we compute this value? Intuition of Bayesian classification:

Use Bayes rule to transform it into a set of other probabilities that are easier to compute

Thomas Bayes: British mathematician (1702-1761)


Bayes Rule

Breaks down any conditional probability P(x|y) into three other probabilities

P(x|y): The conditional probability of an event x assuming that y has occurred


Bayes Rule

We can drop the denominator: it does not change for each tag sequence; we are looking for the best tag sequence for the same observation, for the same fixed set of words


Bayes Rule


Likelihood and prior

n


Likelihood and prior Further Simplifications

n

1. the probability of a word appearing depends only on its own POS tag, i.e, independent of other words around it

2. BIGRAM assumption: the probability of a tag appearing depends only on the previous tag

3. The most probable tag sequence estimated by the bigram tagger



n

1. the probability of a word appearing depends only on its own POS tag, i.e, independent of other words around it

thekoalaputthekeysonthetable

WORDSTAGS

NVPDET



2. BIGRAM assumption: the probability of a tag appearing depends only on the previous tag

Bigrams are groups of two written letters, two syllables, or two words; they are a special case of N-gram.

Bigrams are used as the basis for simple statistical analysis of text

The bigram assumption is related to the first-order Markov assumption



3. The most probable tag sequence estimated by the bigram tagger

n

biagram assumption

---------------------------------------------------------------------------------------------------------------


Two kinds of probabilities (1)

Tag transition probabilities p(ti|ti-1) Determiners likely to precede adjs and nouns

That/DT flight/NN The/DT yellow/JJ hat/NN So we expect P(NN|DT) and P(JJ|DT) to be high But P(DT|JJ) to be:?



Tag transition probabilities p(ti|ti-1) Compute P(NN|DT) by counting in a labeled

corpus:

# of times DT is followed by NN



Word likelihood probabilities p(wi|ti) P(is|VBZ) = probability of VBZ (3sg Pres verb) being “is”

Compute P(is|VBZ) by counting in a labeled corpus:

If we were expecting a third person singular verb, how likely is it that

this verb would be is?


An Example: the verb “race”

Secretariat/NNP is/VBZ expected/VBN to/TO race/VB tomorrow/NR

People/NNS continue/VB to/TO inquire/VB the/DT reason/NN for/IN the/DT race/NN for/IN outer/JJ space/NN

How do we pick the right tag?


Disambiguating “race”


Disambiguating “race”

P(NN|TO) = .00047P(VB|TO) = .83The tag transition probabilities P(NN|TO) and P(VB|TO) answer the question: ‘How likely are we to expect verb/noun given the previous tag TO?’

P(race|NN) = .00057P(race|VB) = .00012Lexical likelihoods from the Brown corpus for ‘race’ given a POS tag NN or VB.

P(NR|VB) = .0027P(NR|NN) = .0012tag sequence probability for the likelihood of an adverb occurring given the previous tag verb or noun

P(VB|TO)P(NR|VB)P(race|VB) = .00000027P(NN|TO)P(NR|NN)P(race|NN)=.00000000032Multiply the lexical likelihoods with the tag sequence probabiliies: the verb wins


Hidden Markov Models

What we’ve described with these two kinds of probabilities is a Hidden Markov Model (HMM)

Let’s just spend a bit of time tying this into the model In order to define HMM, we will first introduce the Markov

Chain, or observable Markov Model.


Definitions

A weighted finite-state automaton adds probabilities to the arcs The sum of the probabilities leaving any arc must sum

to one A Markov chain is a special case of a WFST in which the

input sequence uniquely determines which states the automaton will go through

Markov chains can’t represent inherently ambiguous problems Useful for assigning probabilities to unambiguous

sequences


Markov chain = “First-order observed Markov Model” a set of states

Q = q1, q2…qN; the state at time t is qt a set of transition probabilities:

a set of probabilities A = a01a02…an1…ann. Each aij represents the probability of transitioning from state i to state j The set of these is the transition probability matrix A

Distinguished start and end states

Special initial probability vector

i the probability that the MM will start in state i, each i expresses the probability p(qi|START)

aij P(qt j | qt 1 i) 1i, j N

aij 1; 1i Nj1

N


Markov chain = “First-order observed Markov Model”

Markov Chain for weather: Example 1 three types of weather: sunny, rainy, foggy we want to find the following conditional probabilities:

P(qn|qn-1, qn-2, …, q1)

- I.e., the probability of the unknown weather on day n, depending on the (known) weather of the preceding days

- We could infer this probability from the relative frequency (the statistics) of past observations of weather sequences

Problem: the larger n is, the more observations we must collect.

Suppose that n=6, then we have to collect statistics for 3(6-1) =

243 past histories


Markov chain = “First-order observed Markov Model” Therefore, we make a simplifying assumption, called the (first-order) Markov

assumption

for a sequence of observations q1, … qn,

current state only depends on previous state

the joint probability of certain past and current observations


Markov chain = “First-order observable Markov Model”


Markov chain = “First-order observed Markov Model”

Given that today the weather is sunny, what's the probability that tomorrow is sunny and the day after is rainy?

Using the Markov assumption and the probabilities in table 1, this translates into:


Markov chain for weather

What is the probability of 4 consecutive rainy days? Sequence is rainy-rainy-rainy-rainy I.e., state sequence is 3-3-3-3 P(3,3,3,3) =

1a11a11a11a11 = 0.2 x (0.6)3 = 0.0432


Hidden Markov Model

For Markov chains, the output symbols are the same as the states. See sunny weather: we’re in state sunny

But in part-of-speech tagging (and other things) The output symbols are words But the hidden states are part-of-speech tags

So we need an extension! A Hidden Markov Model is an extension of a Markov

chain in which the output symbols are not the same as the states.

This means we don’t know which state we are in.


Markov chain for words

Observed events: words

Hidden events: tags


States Q = q1, q2…qN; Observations O = o1, o2…oN;

Each observation is a symbol from a vocabulary V = {v1,v2,…vV}

Transition probabilities (prior)

Transition probability matrix A = {aij}

Observation likelihoods (likelihood)

Output probability matrix B={bi(ot)}a set of observation likelihoods, each expressing the probability of an

observation ot being generated from a state i, emission probabilities

Special initial probability vector

i the probability that the HMM will start in state i, each i expresses the probability

p(qi|START)

Hidden Markov Models


Assumptions

Markov assumption: the probability of a particular state depends only on the previous state

Output-independence assumption: the probability of an output observation depends only on the state that produced that observation

P(qi | q1...qi 1) P(qi | qi 1)


HMM for Ice Cream

You are a climatologist in the year 2799 Studying global warming You can’t find any records of the weather in Boston, MA

for summer of 2007 But you find Jason Eisner’s diary Which lists how many ice-creams Jason ate every date

that summer Our job: figure out how hot it was


Noam task

Given Ice Cream Observation Sequence: 1,2,3,2,2,2,3…

(cp. with output symbols) Produce:

Weather Sequence: C,C,H,C,C,C,H …

(cp. with hidden states, causing states)


HMM for ice cream


Different types of HMM structure

Bakis = left-to-right Ergodic = fully-connected


HMM Taggers

Two kinds of probabilities A transition probabilities (PRIOR) B observation likelihoods (LIKELIHOOD)

HMM Taggers choose the tag sequence which maximizes the product of word likelihood and tag sequence probability


Weighted FSM corresponding to hidden states of HMM, showing A probs


B observation likelihoods for POS HMM


The A matrix for the POS HMM


The B matrix for the POS HMM


HMM Taggers

The probabilities are trained on hand-labeled training corpora (training set)

Combine different N-gram levels Evaluated by comparing their output from a test set to

human labels for that test set (Gold Standard)


The Viterbi Algorithm best tag sequence for "John likes to fish in the sea"? efficiently computes the most likely state sequence given a

particular output sequence based on dynamic programming


A smaller example0.6

b

q rstart end

0.5

0.7

What is the best sequence of states for the input string “bbba”?

Computing all possible paths and finding the one with the max probability is exponential

a

0.4 0.80.2

b a

1 1

0.3 0.5


A smaller example (con’t)

For each state, store the most likely sequence that could lead to it (and its probability) Path probability matrix:

An array of states versus time (tags versus words) That stores the prob. of being at each state at each time in terms of the prob. for being

in each state at the preceding time.Best sequence Input sequence / time

ε --> b b --> b bb --> b bbb --> a

leading

to q

coming

from qε --> q 0.6

(1.0x0.6)

q --> q 0.108

(0.6x0.3x0.6)

qq --> q 0.01944 (0.108x0.3x0.6)

qrq --> q 0.018144

(0.1008x0.3x0.4)

coming

from rr --> q 0

(0x0.5x0.6)

qr --> q 0.1008

(0.336x0.5x 0.6)

qrr --> q 0.02688 (0.1344x0.5x0.4)

leading

to r

coming

from qε --> r 0

(0x0.8)

q --> r 0.336

(0.6x0.7x0.8)

qq --> r 0.0648 (0.108x0.7x0.8)

qrq --> r 0.014112

(0.1008x0.7x0.2)

coming

from rr --> r 0 (0x0.5x0.8)

qr --> r 0.1344 (0.336x0.5x0.8)

qrr --> r 0.01344

(0.1344x0.5x0.2)


Viterbi intuition: we are looking for the best ‘path’

promised to back the bill

VBD

VBN

TO

VB

JJ

NN

RB

DT

NNP

VB

NN


VBD

VBN

TO

VB

JJ

NN

RB

DT

NNP

VB

NN

S1 S2 S4S3 S5


VBD

VBN

TO

VB

JJ

NN

RB

DT

NNP

VB

NN

Slide from Dekang Lin


The Viterbi Algorithm


Intuition

The value in each cell is computed by taking the MAX over all paths that lead to this cell.

An extension of a path from state i at time t-1 is computed by multiplying: Previous path probability from previous cell viterbi[t-

1,i] Transition probability aij from previous state I to

current state j Observation likelihood bj(ot) that current state j

matches observation symbol t


Viterbi example


Smoothing of probabilities

Data sparseness is a problem when estimating probabilities based on corpus data. The “add one” smoothing technique –

BN

wCwP n

n

1,1

,1

C- absolute frequencyN: no of training instancesB: no of different types

Linear interpolation methods can compensate for data sparseness with higher order models. A common method is interpolating trigrams, bigrams and unigrams:

iii

iiiiiiii ttPttPtPttP

1,10

)|()|()(| 2,133122111,1

The lambda values are automatically determined using a variant of the Expectation Maximization algorithm.


in bigram POS tagging, we condition a tag only on the preceding tag

why not... use more context (ex. use trigram model)

more precise: “is clearly marked” --> verb, past participle “he clearly marked” --> verb, past tense

combine trigram, bigram, unigram models condition on words too

but with an n-gram approach, this is too costly (too many parameters to model)

Possible improvements


Further issues with Markov Model tagging

Unknown words are a problem since we don’t have the required probabilities. Possible solutions: Assign the word probabilities based on corpus-wide distribution

of POS Use morphological cues (capitalization, suffix) to assign a more

calculated guess. Using higher order Markov models:

Using a trigram model captures more context However, data sparseness is much more of a problem.


TnT

Efficient statistical POS tagger developed by Thorsten Brants, ANLP-2000 Underlying model:

Trigram modelling – The probability of a POS only depends on its two preceding POS The probability of a word appearing at a particular position given that its

POS occurs at that position is independent of everything else.

T

iTTiiiii

ttttPtwPtttP

T 1121 )|()|(),|(maxarg

1


Training

Maximum likelihood estimates:

)(

),()|(:

),(

),,(),|(:

)(

),()|( : Bigrams

: Unigrams

3

3333

32

321213

^

3

3223

33

tc

twctwPLexical

ttc

tttctttPTrigrams

tc

ttcttP

N

)c(t)(tP

^

^

Smoothing : context-independent variant of linear interpolation.

),|()|()(),|( 213

^

323

^

23

^

1213 tttPttPtPtttP


Smoothing algorithm

Set λi=0

For each trigram t1 t2 t3 with f(t1,t2,t3 )>0 Depending on the max of the following three values:

Case (f(t1,t2,t3 )-1)/ f(t1,t2) : incr λ3 by f(t1,t2,t3 )

Case (f(t2,t3 )-1)/ f(t2) : incr λ2 by f(t1,t2,t3 )

Case (f(t3 )-1)/ N-1 : incr λ1 by f(t1,t2,t3 )

Normalize λi


Evaluation of POS taggers

compared with gold-standard of human performance metric:

accuracy = % of tags that are identical to gold standard most taggers ~96-97% accuracy must compare accuracy to:

ceiling (best possible results) how do human annotators score compared to each other? (96-

97%) so systems are not bad at all!

baseline (worst possible results) what if we take the most-likely tag (unigram model) regardless of

previous tags ? (90-91%) so anything less is really bad


More on tagger accuracy is 95% good?

that’s 5 mistakes every 100 words if on average, a sentence is 20 words, that’s 1 mistake per sentence

when comparing tagger accuracy, beware of: size of training corpus

the bigger, the better the results difference between training & testing corpora (genre, domain…)

the closer, the better the results size of tag set

Prediction versus classification unknown words

the more unknown words (not in dictionary), the worst the results


Error Analysis

Look at a confusion matrix (contingency table)

E.g. 4.4% of the total errors caused by mistagging VBD as VBN See what errors are causing problems

Noun (NN) vs ProperNoun (NNP) vs Adj (JJ) Adverb (RB) vs Particle (RP) vs Prep (IN) Preterite (VBD) vs Participle (VBN) vs Adjective (JJ)

ERROR ANALYSIS IS ESSENTIAL!!!


Tag indeterminacy


Major difficulties in POS tagging Unknown words (proper names)

because we do not know the set of tags it can take and knowing this takes you a long way (cf. baseline POS tagger) possible solutions:

assign all possible tags with probabilities distribution identical to lexicon as a whole

use morphological cues to infer possible tags ex. word ending in -ed are likely to be past tense verbs or past participles

Frequently confused tag pairs preposition vs particle

<running> <up> a hill (prep) / <running up> a bill (particle) verb, past tense vs. past participle vs. adjective


Unknown Words

Most-frequent-tag approach. What about words that don’t appear in the training set? Suffix analysis:

The probability distribution for a particular suffix is generated from all words in the training set that share the same suffix.

Suffix estimation – Calculate the probability of a tag t given the last i letters of an n letter word.

Smoothing: successive abstraction through sequences of increasingly more general contexts (i.e., omit more and more characters of the suffix)

Use a morphological analyzer to get the restriction on the possible tags.


Unknown words


Alternative graphical models for part of speech tagging


Different Models for POS tagging

HMM Maximum Entropy Markov Models Conditional Random Fields


Hidden Markov Model (HMM) : Generative Modeling

Source Model PY

Noisy Channel PXY

y x

i

ii yyPP )|()( 1y i

ii yxPP )|()|( yx


Dependency (1st order)

kY1kY

kX

)|( kk YXP

)|( 1kk YYP

1kX

)|( 11 kk YXP

2kX

)|( 22 kk YXP

2kY)|( 21 kk YYP

1kY

1kX

)|( 1 kk YYP

)|( 11 kk YXP


Disadvantage of HMMs (1)

No Rich Feature Information Rich information are required

When xk is complex When data of xk is sparse

Example: POS Tagging How to evaluate Pwk|tk for unknown words wk ? Useful features

Suffix, e.g., -ed, -tion, -ing, etc. Capitalization

Generative Model Parameter estimation: maximize the joint likelihood of training examples

T

P),(

2 ),(logyx

yYxX


Generative Models

Hidden Markov models (HMMs) and stochastic grammars Assign a joint probability to paired observation and label sequences The parameters typically trained to maximize the joint likelihood of train examples


Generative Models (cont’d)

Difficulties and disadvantages Need to enumerate all possible observation sequences Not practical to represent multiple interacting features or long-range

dependencies of the observations Very strict independence assumptions on the observations


Better Approach Discriminative model which models P(y|x) directly Maximize the conditional likelihood of training examples

T

P),(

2 )|(logyx

xXyY


Maximum Entropy modeling

N-gram model : probabilities depend on the previous few tokens. We may identify a more heterogeneous set of features which contribute in some way

to the choice of the current word. (whether it is the first word in a story, whether the next word is to, whether one of the last 5 words is a preposition, etc)

Maxent combines these features in a probabilistic model. The given features provide a constraint on the model. We would like to have a probability distribution which, outside of these constraints, is

as uniform as possible – has the maximum entropy among all models that satisfy these constraints.


Maximum Entropy Markov Model Discriminative Sub Models

Unify two parameters in generative model into one conditional model

Two parameters in generative model,

parameter in source model and parameter in

noisy channel

Unified conditional model Employ maximum entropy principle

)|( 1kk yyP

)|( kk yxP

),|( 1kkk yxyP

i

iii xyyPP ),|()|( 1xy

Maximum Entropy Markov Model


General Maximum Entropy Principle

Model Model distribution PY|X with a set of features

fffl defined on X and Y

Idea Collect information of features from training data Principle

Model what is known Assume nothing else

Flattest distribution

Distribution with the maximum Entropy


Example

(Berger et al., 1996) example Model translation of word “in” from English to French

Need to model P(wordFrench) Constraints

1: Possible translations: dans, en, à, au course de, pendant 2: “dans” or “en” used in 30% of the time 3: “dans” or “à” in 50% of the time


Features

Features 0-1 indicator functions

1 if x y satisfies a predefined condition 0 if not

Example: POS Tagging

otherwise

NN is and tion- with ends if

,0

,1),(1

yxyxf

otherwise ,0

NNP is andtion Captializa with starts if ,1),(2

yxyxf


Constraints

Empirical Information Statistics from training data T

Tyx

ii yxfT

fP),(

),(||

1)(ˆ

Constraints)()(ˆ

ii fPfP

Tyx YDy

ii yxfxXyYPT

fP),( )(

),()|(||

1)(

Expected Value From the distribution PY|X we want to model


Maximum Entropy: Objective

Entropy

x y

Tyx

xXyYPxXyYPxP

xXyYPxXyYPT

I

)|(log)|()(ˆ

)|(log)|(||

1

2

),(2

)()(ˆ s.t.

max)|(

fPfP

IXYP

Maximization Problem


Dual Problem

Dual Problem Conditional model

Maximum likelihood of conditional data)),(exp()|(

1

l

iii yxfxXyYP

Solution Improved iterative scaling (IIS) (Berger et al. 1996) Generalized iterative scaling (GIS) (McCallum et al.

2000)

Tyx

xXyYPl ),(

2,,

)|(logmax1


Maximum Entropy Markov Model Use Maximum Entropy Approach to Model

1st order

),|( 11 kkkkkk yYxXyYP

Features Basic features (like parameters in HMM)

Bigram (1st order) or trigram (2nd order) in source model

State-output pair feature Xkxk Yk yk Advantage: incorporate other advanced

features on xk yk

HMM vs MEMM (1st order)

kY1kY

kX

)|( 1kk YYP

)|( kk YXP

HMMMaximum Entropy

Markov Model (MEMM)

kY1kY

kX

),|( 1kkk YXYP


Performance in POS Tagging

POS Tagging Data set: WSJ Features:

HMM features, spelling features (like –ed, -tion, -s, -ing, etc.)

Results (Lafferty et al. 2001) 1st order HMM

94.31% accuracy, 54.01% OOV accuracy 1st order MEMM

95.19% accuracy, 73.01% OOV accuracy


ME applications

Part of Speech (POS) Tagging (Ratnaparkhi, 1996) P(POS tag | context) Information sources

Word window (4) Word features (prefix, suffix, capitalization) Previous POS tags


ME applications

Abbreviation expansion (Pakhomov, 2002) Information sources

Word window (4) Document title

Word Sense Disambiguation (WSD) (Chao & Dyer, 2002) Information sources

Word window (4) Structurally related words (4)

Sentence Boundary Detection (Reynar & Ratnaparkhi, 1997) Information sources

Token features (prefix, suffix, capitalization, abbreviation) Word window (2)


Solution

Global Optimization Optimize parameters in a global model simultaneously,

not in sub models separately Alternatives

Conditional random fields Application of perceptron algorithm


Why ME?

Advantages Combine multiple knowledge sources

Local Word prefix, suffix, capitalization (POS - (Ratnaparkhi, 1996)) Word POS, POS class, suffix (WSD - (Chao & Dyer, 2002)) Token prefix, suffix, capitalization, abbreviation (Sentence Boundary -

(Reynar & Ratnaparkhi, 1997)) Global

N-grams (Rosenfeld, 1997) Word window Document title (Pakhomov, 2002) Structurally related words (Chao & Dyer, 2002) Sentence length, conventional lexicon (Och & Ney, 2002)

Combine dependent knowledge sources


Why ME?

Advantages Add additional knowledge sources Implicit smoothing

Disadvantages Computational

Expected value at each iteration Normalizing constant

Overfitting Feature selection

Cutoffs Basic Feature Selection (Berger et al., 1996)


Conditional Models

Conditional probability P(label sequence y | observation sequence x) rather than joint probability P(y, x) Specify the probability of possible label sequences given an observation

sequence

Allow arbitrary, non-independent features on the observation sequence X

The probability of a transition between labels may depend on past and future observations Relax strong independence assumptions in generative models


Discriminative ModelsMaximum Entropy Markov Models (MEMMs)

Exponential model Given training set X with label sequence Y:

Train a model θ that maximizes P(Y|X, θ) For a new data sequence x, the predicted label y maximizes P(y|x, θ) Notice the per-state normalization


MEMMs (cont’d)

MEMMs have all the advantages of Conditional Models

Per-state normalization: all the mass that arrives at a state must be distributed among the possible successor states (“conservation of score mass”)

Subject to Label Bias Problem

Bias toward states with fewer outgoing transitions


Label Bias Problem

• P(1 and 2 | ro) = P(2 | 1 and ro)P(1 | ro) = P(2 | 1 and o)P(1 | r) P(1 and 2 | ri) = P(2 | 1 and ri)P(1 | ri) = P(2 | 1 and i)P(1 | r)

• Since P(2 | 1 and x) = 1 for all x, P(1 and 2 | ro) = P(1 and 2 | ri)In the training data, label value 2 is the only label value observed after label value 1Therefore P(2 | 1) = 1, so P(2 | 1 and x) = 1 for all x

• However, we expect P(1 and 2 | ri) to be greater than P(1 and 2 | ro).

• Per-state normalization does not allow the required expectation

• Consider this MEMM:


Solve the Label Bias Problem

Change the state-transition structure of the model

Not always practical to change the set of states

Start with a fully-connected model and let the training procedure figure out a good structure Prelude the use of prior, which is very valuable (e.g. in information extraction)


Random Field


Conditional Random Fields (CRFs)

CRFs have all the advantages of MEMMs without label bias problem MEMM uses per-state exponential model for the conditional probabilities

of next states given the current state CRF has a single exponential model for the joint probability of the entire

sequence of labels given the observation sequence Undirected acyclic graph Allow some transitions “vote” more strongly than others depending on the

corresponding observations


Definition of CRFs

X is a random variable over data sequences to be labeled

Y is a random variable over corresponding label sequences


Example of CRFs


Graphical comparison among HMMs, MEMMs and CRFs

HMM MEMM CRF


Conditional Distribution

1 2 1 2( , , , ; , , , ); andn n k k

x is a data sequencey is a label sequence v is a vertex from vertex set V = set of label random variablese is an edge from edge set E over Vfk and gk are given and fixed. gk is a Boolean vertex feature; fk is a

Boolean edge featurek is the number of features

are parameters to be estimated

y|e is the set of components of y defined by edge ey|v is the set of components of y defined by vertex v

If the graph G = (V, E) of Y is a tree, the conditional distribution over the label sequence Y = y, given X = x, by fundamental theorem of random fields is:

(y | x) exp ( , y | , x) ( , y | , x)

k k e k k v

e E,k v V ,k

p f e g v


Conditional Distribution (cont’d)

• CRFs use the observation-dependent normalization Z(x) for the conditional distributions:

Z(x) is a normalization over the data sequence x

(y | x) exp ( , y | , x) ( , y |1

(x), x)

k k e k k v

e E,k v V ,k

p f e g vZ


Parameter Estimation for CRFs

The paper provided iterative scaling algorithms

It turns out to be very inefficient

Prof. Dietterich’s group applied Gradient Descendent Algorithm, which is quite efficient


Training of CRFs (From Prof. Dietterich)

log ( | )( , y | , x) ( , y | , x) log (x)

k k e k k ve E,k v V ,k

p y xf e g v Z

log ( | ) ( , y | , x) ( , y | , x) log (x)k k e k k ve E,k v V ,k

p y x f e g v Z

• First, we take the log of the equation

• Then, take the derivative of the above equation

• For training, the first 2 items are easy to get. • For example, for each k, fk is a sequence of Boolean numbers, such

as 00101110100111. is just the total number of 1’s in the sequence.( , y | , x)k k ef e

• The hardest thing is how to calculate Z(x)


Training of CRFs (From Prof. Dietterich) (cont’d)

• Maximal cliques

y1 y2 y3 y4c1 c2 c3

c1 c2 c3

1 2 3 4

1 2 3 4

1 1 2 2 2 3 3 3 4y ,y ,y ,y

1 1 2 2 2 3 3 3 4y y y y

(x) (y ,y ,x) (y ,y ,x) (y ,y ,x)

(y ,y ,x) (y ,y ,x) (y ,y ,x)

Z c c c

c c c

3 4 3 4 3 3 4: exp( (y ,x) (y ,y ,x)) (y ,y ,x)c c

1 1 2 1 2 1 1 2: exp( (y ,x) (y ,x) (y ,y ,x)) (y ,y ,x)c c

2 3 2 3 2 2 3: exp( (y ,x) (y ,y ,x)) (y ,y ,x)c c


POS tagging Experiments


POS tagging Experiments (cont’d)

• Compared HMMs, MEMMs, and CRFs on Penn treebank POS tagging• Each word in a given input sentence must be labeled with one of 45 syntactic tags• Add a small set of orthographic features: whether a spelling begins with a number

or upper case letter, whether it contains a hyphen, and if it contains one of the following suffixes: -ing, -ogy, -ed, -s, -ly, -ion, -tion, -ity, -ies

• oov = out-of-vocabulary (not observed in the training set)


Summary

Discriminative models are prone to the label bias problem

CRFs provide the benefits of discriminative models

CRFs solve the label bias problem well, and demonstrate good performance

Date post:	16-Jan-2016
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CS60057 Speech &Natural Language Processing

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