comp 791a: statistical language processing

1

COMP 791A: Statistical Language Processing

Mathematical EssentialsChap. 2

2

Motivations Statistical NLP aims to do statistical inference

for the field of NL

Statistical inference consists of: taking some data (generated in accordance with

some unknown probability distribution) then making some inference about this distribution

Ex. of statistical inference: language modeling how to predict the next word given the previous

words to do this, we need a model of the language probability theory helps us finding such model

3

Notions of Probability Theory

Probability theory deals with predicting how likely it is that

something will happen Experiment (or trial)

the process by which an observation is made Ex. tossing a coin twice

4

Sample Spaces and events Sample space Ω :

set of all possible basic outcomes of an experiment Coin toss: Ω = head, tail Tossing a coin twice: Ω = HH, HT, TH, TT Uttering a word: |Ω| = vocabulary size

Every observation (element in Ω) is a basic outcome or sample point

An event A is a set of basic outcomes with A Ω Ω is then the certain event Ø is the impossible (or null) event

Example - rolling a die: Sample space Ω = 1, 2, 3, 4, 5, 6 Event A that an even number occurs A = 2, 4, 6

5

Events and Probability

The probability of an event A is denoted p(A) also called the prior probability i.e. the probability before we consider any additional

knowledge Example: experiment of tossing a coin 3 times

Ω = HHH, HHT, HTH, HTT, THH, THT, TTH, TTT events with two or more tails:

A = HTT, THT, TTH, TTT P(A) = |A|/|Ω| = ½ (assuming uniform distribution)

events with all heads: A = HHH P(A) = |A|/|Ω| = ⅛

6

Probability Properties A probability function P (or probability distribution):

Distributes a probability mass of 1 over the sample space Ω [0,1] P(Ω) = 1 For disjoint events Ai (ie : AiAj = Ø for all i ≠j)

P( Ai) = Σ P(Ai)

Immediate consequences: P(Ø ) = 0 P(Ā) = 1 - P(A) AB ==> P(A) ≤ P(B) ΣaєΩ P(a) = 1

7

Joint probability

Joint probability of A and B: P(A,B) = P(AB)

Ω

A BA

B

8

Conditional probability Prior (or unconditional) probability

Probability of an event before any evidence is obtained P(A) = 0.1 P(rain today) = 0.1 i.e. Your belief about A given that you have no evidence

Posterior (or conditional) probability Probability of an event given that all we know is B (some

evidence) P(A|B) = 0.8 P(rain today| cloudy) = 0.8 i.e. Your belief about A given that all you know is B

9

Conditional probability (con’t)

Ω

A BA

B

P(B)B)P(A,

B)|P(A

10

Chain rule

With 3 events, the probability that A, B and C occur is: The probability that A occurs Times, the probability that B occurs, assuming that A

occurred Times, the probability that C occurs, assuming that A and B

have occurred With multiple events, we can generalize to the Chain rule:

P(A1, A2, A3, A4, ..., An)

= P (Ai)

= P(A1) × P(A2|A1) × P(A3|A1,A2) × ... × P(An|A1,A2,A3,…,An-1)

(important to NLP)

P(B) x B)|P(A B)P(A, so P(B)

B)P(A, B)|P(A P(A) x A)|P(B B)P(A, so

P(A)B)P(A,

A)|P(B

13

Example Rare syntactic construction occurs in

1/100,000 sentences

A system identifies sentences with such a construction, but it is not perfect If sentence has the construction -->

system identifies it 95% of the time If sentence does not have the construction -->

system says it does 0.5% of the time

Question: if the system says that sentence S has the

construction… what is the probability that it is right?

14

Example (con’t) What is P(sentence has the construction | the system says yes) ? Let:

cons = sentence has the construction yes = system says yes not_cons = sentence does not have the construction

we have: P(cons) = 1/100,000 = 0.00001 P(yes | cons) = 95% = 0.95 P(yes | not_cons) = 0.5% = 0.005

P(yes) = ? P(B) = P(B|A) P(A) + P(B|Ā) P(Ā) P(yes) = P(yes | cons) × P(cons) + P(yes | not_cons) × P(not_cons)

= 0.95 × 0.00001 + 0.005 × 0.9999

P(yes)P(cons)cons)|P(yes

yes)|P(cons

15

Example (con’t)

(0.2%) 0.002 0.99990.005 0.000010.95

0.000010.95

P(yes)P(cons)cons)|P(yes

yes)|P(cons

So in only 1 sentence out of 500 that the system says yes, it is actually right!!!

So:

16

How likely are we to have Head in a coin toss, given that it is raining today?

A: having a head in a coin toss B: raining today Some variables are independent…

How likely is the word “ambulance” to appear, given that we’ve seen “car accident”?

Words in text are not independent

Statistical Independence vs. Statistical Dependence

17

Independent events Two events A and B are independent:

if the occurrence of one of them does not influence the occurrence of the other

i.e. A is independent of B if P(A) = P(A|B)

If A and B are independent, then: P(A,B) = P(A|B) x P(B) (by chain rule)

= P(A) x P(B) (by independence)

In NLP, we often assume independence of variables

18

Bayes’ Theorem revisited (a golden rule in statistical NLP) If we are interested in which event B is most likely to occur

given an observation A we can chose the B with the largest P(B|A)

P(A) is a normalization constant (to ensure 0…1) is the same for all possible Bs (and is hard to gather anyways) so we can drop it

So Bayesian reasoning:

In NLP:

P(A)P(B) x B)|P(A

argmax A)|P(B argmax BB

_event)P(language vent)language_e|ionP(observatargmax ventlanguage_e

P(B) x B)|P(A argmaxB

19

Application of Bayesian Reasoning Diagnostic systems:

P(Disease | Symptoms) Categorization:

P(Category of object| Features of object) Text classification: P(sports-news | words in text) Character recognition: P(character | bitmap) Speech recognition: P(words | signals) Image processing: P(face-person | image) …

20

Random Variables A random variable X is a function

X: Ω --> Rn (typically n= 1)

Example – tossing 2 dice Ω = (1,1), (1,2), (1,3), … (6,6) X : Ω --> Rx assigns to each point in Ω, the sum of the 2

dice

X(1,1) = 2 X(1,2) = 3, … X(6,6) = 12 Rx= 2,3,4,5,6,7,8,9,10,11,12

A random variable X is discrete if: X: Ω --> S where S is a countable subset of R In particular, if X: Ω --> 0,1

then X is called a Bernoulli trial. A random variable X is continuous if:

X: Ω --> S where S is a continuum of numbers

21

Probability distribution of an RV Let X be a finite random variable

Rx= x1, x2, x3,… xn

A probability mass function f gives the probability of X at different in points in Rx

f(xk) = P(X=xk) = p( xk) p(xk) ≥ 0 Σk p(xk) = 1

X x1 x2 x3 … xn

p(X) p(x1 ) p(x2 ) p(x3 ) … p(xn )

22

Example: Tossing 2 dice X = sum of the faces

X: Ω --> S Ω = (1,1), (1,2), (1,3), …, (6,6) S = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

X = maximum of the faces X: Ω --> S Ω = (1,1), (1,2), (1,3), …, (6,6) S = 1, 2, 3, 4, 5, 6

X 2 3 4 5 6 7 8 9 10 11 12

p(X) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

X 1 2 3 4 5 6

p(X) 1/36 3/36 5/36 7/36 9/36 11/36

23

Expectation The expectation (μ) is the mean (or average or

expected value) of a random variable X

Intuitively, it is: the weighted average of the outcomes where each outcome is weighted by its probability

ex: the average sum of the dice

If X and Y are 2 random variables on the same sample space, then: E(X+Y) = E(X) + E(Y)

ii x )p(xE(X)

24

Example The expectation of the sum of the faces on two dice?

(the average sum of the dice) If equiprobable… (2+3+4+5+…+12)/11 But not, equiprobable

Or more simply: E(SUM)=E(Die1+Die2)=E(Die1)+E(Die2) Each face on 1 die is equiprobable

E(Die) = (1+2+3+4+5+6)/6 = 3.5E(SUM) = 3.5 + 3.5 = 7

736252

361

12362

11363

4362

3361

2)p(xxE(X) ii

...

SUM (xi) 2 3 4 5 6 7 8 9 10 11 12

p(SUM=xi) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

25

Variance and standard deviation The variance of a random variable X is a

measure of whether the values of the RV tend to be consistent over trials or to vary a lot

The standard deviation of X is the square root of the variance

Both measure the weighted “spread” of the values xi around the mean E(X)

2

ii E(X))-(x )p(xvar(X)

var(X)σx

26

Example

What is the variance of the sum of the faces on two dice?

5.837)(12361

7)(11362

7)(6365

7)(5364

7)(4363

7)(3362

7)(2361

E(SUM))-(x )p(xvar(SUM)

222

2222

2

ii

...

SUM (xi) 2 3 4 5 6 7 8 9 10 11 12

p(SUM=xi) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

27

Back to NLP

What is the probability that someone says the sentence:“Mary is reading a book.”

In general, for language events, the probability function P is unknown

We need to estimate P (or a model M of the language) by looking at a sample of data (training set)

2 approaches: Frequentist statistics Bayesian statistics (we will not see)

Language event

x1 x2 x3

p ? ? ?

28

Frequentist Statistics

To estimate P, we use the relative frequency of the outcome in a sample of data

i.e. the proportion of times a certain outcome o occurs.

Where C(o) is the number of times o occurs in N trials For N--> ∞ the relative frequency stabilizes to some

number: the estimate of the probability function

Two approaches to estimate the probability function:

Parametric (assuming a known distribution) Non-parametric (distribution free)… we will not see

NC(o)

fo

29

Parametric Methods

Assume that some phenomenon in language is modeled by a well-known family of distributions (ex. binomial, normal)

The advantages: we have an explicit probabilistic model of the

process by which the data was generated determining a particular probability distribution

within the family requires only the specification of a few parameters (so, less training data)

But: Our assumption on the probability distribution may

be wrong…

30

Non-Parametric Methods

No assumption is made about the underlying distribution of the data

For ex, we can simply estimate P empirically by counting a large number of random events

But: because we use less prior information (no assumption on the distribution), more training data is needed

31

Standard Distributions

Many applications give rise to the same basic form of a probability distribution - but with different parameters.

Discrete Distributions: the binomial distribution (2 outcomes) the multinomial distribution (more than 2 outcomes) …

Continuous Distributions: the normal distribution (Gaussian) …

32

Binomial Distribution (discrete) Also known as Bernoulli distribution

Each trial has only two outcomes (success or failure) The probability of success is the same for each trial The trials are independent There are a fixed number of trials

Distribution has 2 parameters: nb of trials n probability of success p in 1 trial

Ex: Flipping a coin 10 times and counting the number of heads that occur

Can only get a head or a tail (2 outcomes) For each flip there is the same chance of getting a head (same

prob.) The coin flips do not effect each other (independence) There are 10 coin flips (n = 10)

33

Examples

b(n,p) = B(10, 0.7)

Nb trials = 10 Prob(head) = 0.7

b(n,p) = B(10, 0.1)

Nb trials = 10 Prob(head) = 0.1

34

Binomial probability function let:

n = nb of trials p = probability of success in any trial r = nb of successes out of the n trials

n)r(0r!r)!(n

n!r

n:where

p1 p r

n

trials n in successes r having of yprobabilit theP(r)

rnr

The number of ways of having r successes in n trials.

The probability of having r successes

The probability of having n-r failures.

35

Example

What is the probability of rolling higher than 4 in 2 rolls of 3 dice rolls?

1st 2nd 3rd Probability

4 4 44 4 4

64

64

64

62

64

64

44

44

4

4

4

44 4

444 44 4 4

64

62

64

62

62

64

64

64

62

62

64

62

64

62

62

62

62

62

*

*

*

92

272

272

272

64

62

62

62

64

62

62

62

64 )()()(2)p(r

92

64

364

642

62 3

2

32)p(r

n trials =3p probability of success in 1 trial = r successes = 2

4

62

36

Properties of binomial distribution

B(n,p) Mean E(X) = μ = np

Ex: Flipping a coin 10 times E(head) = 10 x ½ = 5

Variance σ2= np(1-p) Ex:

Flipping a coin 10 times σ2 = 10 x ½ ( ½ ) = 2.5

37

Binomial distribution in NLP Works well for tossing a coin But, in NLP we do not always have complete independence from

one trial to the next Consecutive sentences are not independent Consecutive POS tags are not independent

So, binomial distribution in NLP is an approximation (but a fair one)

When we count how many times something is present or absent And we ignore the possibility of dependencies between one trial

and the next Then, we implicitly use the binomial distribution

Ex: Count how many sentences contain the word “the” Assume each sentence is independent

Count how many times a verb is used as transitive Assume each occurrence of the verb is independent of the others…

38

Also known as Gaussian distribution (or Bell curve) to model a random variable X on an infinite sample space (ex. height, length…)

X is a continuous random variable if there is a function f(x) defined on the real line R = (-∞, +∞) such that:

f is non-negative f(x) ≥ 0 The area under the curve of f is one The probability that X lies in the interval [a,b] is equal to

the area under f between x=a and x=b

Normal Distribution (continuous)

1dx f(x)

b

adx f(x)b)XP(a

39

has 2 parameters: mean μ standard deviation σ

Normal Distribution (con’t)

2

2

2σ

μ)(x

e2πσ

1p(x)

n(μ,σ)= n(0,1)

μ=0; σ= 1

n(μ,σ)=n(1.5,2)

μ=1.5; σ=2

40

The standard normal distribution if μ=0 and σ=1, then called standard normal

distribution Z

1

034.1%dx f(x)1)XP(0

41

Frequentist vs Bayesian Statistics

Assume we toss a coin 10 times, and get 8 heads: Frequentists will conclude (from the observations) that a

head comes 8/10 -- Maximum Likelihood Estimate (MLE) if we look at the coin, we would be reluctant to accept

8/10… because we have prior beliefs

Bayesian statisticians will use an a-priori probability distribution (their belief)

will update the beliefs when new evidence comes in (a sequence of observations)

by calculating the Maximum A Posteriori (MAP) distribution. The MAP probability becomes the new prior probability and

the process repeats on each new observation

42

Essential Information Theory

Developed by Shannon in the 40s To maximize the amount of information

that can be transmitted over an imperfect communication channel (the noisy channel)

Notion of entropy (informational content): How informative is a piece of information?

ex. How informative is the answer to a question If you already have a good guess about the answer, the

actual answer is less informative… low entropy

43

Entropy - intuition Ex: Betting 1$ to the flip of a coin

If the coin is fair: Expected gain is ½ (+1) + ½ (-1) = 0$ So you’d be willing to pay up to 1$ for advanced information

(1$ - 0$ average win)

If the coin is rigged P(head) = 0.99 P(tail) = 0.01

assuming you bet on head (!) Expected gain is 0.99(+1) + 0.01(-1) = 0.98$ So you’d be willing to pay up to 2¢ for advanced information

(1$ - 0.98$ average win)

Entropy of fair coin is 1$ > entropy of rigged coin 0.02$

44

Entropy

Let X be a discrete RV Entropy (or self-information)

measures the amount of information in a RV average uncertainty of a RV the average length of the message needed to transmit

an outcome xi of that variable the size of the search space consisting of the possible

values of a RV and its associated probabilities measured in bits Properties:

H(X) ≥ 0 If H(X) = 0 then it provides no new information

n

1ii2i )p(x)logp(xH(X)

45

Example: The coin flip Fair coin:

Rigged coin:

bit 121

log21

21

log21

- )p(x)logp(xH(X) 22

n

1ii2i

bits 0.081001

log1001

10099

log10099

- )p(x)logp(xH(X) 22

n

1ii2i

P(head)

Entr

op

y

46

In simplified Polynesian, we have 6 letters with frequencies:

The per-letter entropy is

We can design a code that on average takes 2.5bits to transmit a letter

Can be viewed as the average nb of yes/no questions you need to ask to identify the outcome (ex: is it a ‘t’? Is it a ‘p’?)

p t k a i u

1/8 1/4 1/8 1/4 1/8 1/8

bits 2.5)81

log81

81

log81

41

log41

81

log81

41

log41

81

log81

(p(i)p(i)logH(p) 222222ui,a,k,t,p,i

2

p t k a i u

100 00 101

01 110

111

Example: Simplified Polynesian

47

Entropy in NLP

Entropy is a measure of uncertainty The more we know about something the lower its

entropy So if a language model captures more of the

structure of the language, then its entropy should be lower

in NLP, language models are compared by using their entropy. ex: given 2 grammars and a corpus, we use entropy

to determine which grammar beter matches the corpus.

comp 791a: statistical language processing

Documents