Transcript Basic statistics and n

BASIC NOTIONS OF
PROBABILITY THEORY
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What probability theory is for
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Suppose you’ve already texted the characters
“There in a minu”
You’d like your mobile phone to guess the most likely
completion of “minu” rather than MINUET or MINUS or
MINUSCULE
In other words, you’d like your mobile phone to know
that given what you’ve texted so far, MINUTE is more
likely than those other alternatives
PROBABILITY THEORY was developed to formalize
the notion of LIKELIHOOD
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TRIALS (or EXPERIMENTS)
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Anything that may have a certain OUTCOME
(on which you can make a bet, say)
Classic examples:
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In NLE:
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Throwing a die
A horse race
Looking at the next word in a text
Having your NL system perform a certain task
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(ELEMENTARY) OUTCOMES
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The results of an experiment:
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In a coin toss, HEAD or TAILS
In a race, the names of the horses involved
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In NLE:
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Or if we are only interested in whether a particular horse
wins: WIN and LOSE
When looking at the next word: the possible words
In the case of a system: RIGHT or WRONG
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EVENTS
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Often, we want to talk about the likelihood of
getting one of several outcomes:
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An EVENT is a set of possible OUTCOMES
(possibly just a single elementary outcome):
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E.g., with dice, the likelihood of getting an even
number, or a number greater than 3
E1 = {4}
E2 = {2,4,6}
E3 = {3,4,5,6}
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SAMPLE SPACES
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The SAMPLE SPACE is the set of all possible
outcomes:
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For the case of a dice, sample space S = {1,2,3,4,5,6}
For the texting case:
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Texting a word is a TRIAL,
The word texted is an OUTCOME,
EVENTS which result from this trial are: texting the word
“minute”, texting a word that begins with “minu”, etc
The set of all possible words is the SAMPLE SPACE
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(NB: the sample space may be very large, or even infinite)
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Probability Functions
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The likelihood of an event is indicated using a
PROBABILITY FUNCTION P
The probability of an event E is specified by a function
P(E), with values between 0 and 1
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Example: in the case of die casting,
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P(E) = 1: the event is CERTAIN to occur
P(E) = 0: the event is certain NOT to occur
P(E’ = ‘getting as a result a number between 1 and 6’) =
P({1,2,3,4,5,6}) = 1
P(E’’ = ‘getting as a result 7’) = 0
The sum of the probabilities of all elementary
outcomes = 1
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Probabilities and
relative frequencies
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In the case of a die, we know all of the possible outcomes ahead
of time, and we also know a priori what the likelihood of a certain
outcome is. But in many other situations in which we would like to
estimate the likelihood of an event, this is not the case.
For example, suppose that we would like to bet on horses rather
than on dice. Harry is a race horse: we do not know ahead of time
how likely it is for Harry to win. The best we can do is to
ESTIMATE P(WIN) using the RELATIVE FREQUENCY of the
outcome `Harry wins’
Suppose Harry raced 100 times, and won 20 races overall. Then
– P(WIN) = WIN/TOTAL NUMBER OF RACES = .2
– P(LOSE) = .8
The use of probabilities we are interested in (estimate the
probability of certain sequences
of words) is of this type
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Joint probabilities
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We are often interested in the probability of TWO events
happening:
– When throwing a die TWICE, the probability of getting a 6 both
times
– The probability of finding a sequence of two words: `the’ and
`car’
We use the notation A&B to indicate the conjunction of two events,
and P(A&B) to indicate the probability of such conjunction
– Because events are SETS, the probability is often also written
as
P( A  B )
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We use the same notation with WORDS: P(‘the’ & ‘car’)
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Other combinations of events
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A  B: either event A or event B happens
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 A: event A does not happen
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P(A  B) = P(A) + P(B) – P(AB)
P( A) = 1 –P(A)
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Prior probability vs. conditional
probability
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The prior probability P(WIN) is the likelihood of an event occurring
irrespective of anything else we know about the world
Often however we DO have additional information, that can help
us making a more informed guess about the likelihood of a certain
event
E.g, take again the case of Harry the horse. Suppose we know
that it was raining during 30 of the races that Harry raced, and that
Harry won 15 of these races. Intuitively, the probability of Harry
winning when it’s raining is .5 - HIGHER than the probability of
Harry winning overall
– We can make a more informed guess
We indicate the probability of an event A happening given that we
know that event B happened as well – the CONDITIONAL
PROBABILITY of A given B – as P(A|B)
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Conditional probability
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Conditional probability is DEFINED as follows:
P( A & B)
P( A | B) 
P( B)
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Intuitively, you RESTRICT the range of trials in
consideration to those in which event B took place, as
well (most easily seen when thinking in terms of
relative frequency)
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Conditional probability
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Example
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Consider the case of Harry the horse again:
P(WIN & RAIN)
P(WIN | RAIN) 
P( RAIN)
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Where:
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P(WIN&RAIN) = 15/100 = .15
P(RAIN) = 30/100 = .30
This gives:
0.15
P (WIN | RAIN ) 
 0.5
0.3
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(in agreement with our intuitions)
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The chain rule
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The definition of conditional probability can we
rewritten as:
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These equation generalize to the so-called CHAIN
RULE:
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P(w1,w2,w3,….wn) = P(w1) P(w2|w1) P(w3|w1,w2) …. P(wn|w1 ….
wn-1)
The chain rule plays an important role in statistical
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P(A&B) = P(A|B) P(B)
P(A&B) = P(B|A) P(A)
P(the big dog) = P(the) P(big|the) P(dog|the big)
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Independence
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Additional information does not always help. For example,
knowing the color of a dice usually doesn’t help us predicting the
result of a throw; knowing the name of the jockey’s girlfriend
doesn’t help predicting how well the horse he rides will do in a
race; etc. When this is the case, we say that two events are
INDEPENDENT
The notion of independence is defined in probability theory using
the definition of conditional probability
Consider again the basic form of the chain rule:
– P(A&B) = P(A|B) P(B)
We say that two events are INDEPENDENT if:
– P(A&B) = P(A) P(B)
– P(A|B) = P(A)
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Bayes’ theorem
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Suppose you’ve developed an IR system for searching
a big database (say, the Web)
Given any search, about 1/100,000 documents is
relevant (REL)
Suppose your system is pretty good:
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What is the probability that the document is relevant,
when the system says YES?
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P(YES|REL) = .95
P(YES| REL) = .005
P(YES|REL)?
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Bayes’ Theorem
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Bayes’ Theorem is a pretty trivial consequence of the
definition of conditional probability, but it is very useful
in that it allows us to use one conditional probability to
compute another
We already saw that the definition of conditional
probability can be rewritten equivalently as:
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P(A&B) = P(A|B) P(B)
P(A&B) = P(B|A) P(A)
If we equate the two left sides, we get Bayes’ theorem
P( A | B) P( B)
P( B | A) 
P( A)
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Application of Bayes’ theorem
P(YES | REL) P( REL)
P( REL | YES) 
P(YES)
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P(YES | REL) P( REL)
P(YES | REL) P( REL)  P(YES | REL) P(REL)
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0.95  0.00001
 0.002
0.95  0.00001  0.005  0.99999
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Statistical NLE
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What’s the connection between this and natural language?
A number of NL interpretation (and generation) tasks can be formulated in
terms of CHOICE BETWEEN ALTERNATIVES: choosing the most likely
– continuation of a certain sentence
– POS tag or meaning for a word
– Parse for a sentence
In all of these cases, we can formalize `likelihood’ using probabilities, and
choose the alternative with THE HIGHEST PROBABILITY
Tomorrow we will see the first (and simplest) example of this: choosing
the most likely next word
This task can be viewed as the task of choosing the w that maximizes:
P(w | W1 …. WN-1)
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Using corpora to estimate
probabilities
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But where do we get these probabilities? Idea:
estimate them by RELATIVE FREQUENCY.
The simplest method: Maximum Likelihood
Estimate (MLE). Count the number of words in
a corpus, then count how many times a given
sequence is encountered.
C (W1..Wn )
P (W1..Wn ) 
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Readings
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Krenn and Samuelsson, The Linguist’s Guide
to Statistics (on the Web site)
The Statistics Glossary
Further reading: Manning and Schuetze,
chapter 2.1
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