INTRODUCTION TO ARTIFICIAL INTELLIGENCE

Massimo Poesio

LECTURE 11 (Lab): Probability reminder

MACHINE LEARNING

• The next series of lectures will be on methods to learn how to solve problems from data (MACHINE LEARNING)

• Most methods of this type presuppose some knowledge of probability and statistics

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WHY PROBABILITY THEORY

• 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)

• A trial is anything that may have a certain OUTCOME (on which you can make a bet, say)

• Classic examples:– Throwing a die (outcomes: 1, 2, 3, 4, 5 , 6)– A horse race (outcomes?)

• In NLE:– Looking at the next word in a text– Having your NL system perform a certain task

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(ELEMENTARY) OUTCOMES

• The results of an experiment:– In a coin toss, HEAD or TAILS– In a race, the names of the horses involved• Or if we are only interested in whether a particular

horse wins: WIN and LOSE

• In NLE:– When looking at the next word: the possible

words– In the case of a system: RIGHT or WRONG

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EVENTS

• Often, we want to talk about the likelihood of getting one of several outcomes:– E.g., with dice, the likelihood of getting an even

number, or a number greater than 3• An EVENT is a set of possible OUTCOMES

(possibly just a single elementary outcome):– E1 = {4}– E2 = {2,4,6}– E3 = {3,4,5,6}

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SAMPLE SPACES

• The SAMPLE SPACE is the set of all possible outcomes:– For the case of a dice, sample space S = {1,2,3,4,5,6}– For the case of a coin toss, sample space S = {H,T}

• For the texting case:– 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

• (NB: the sample space may be very large, or even infinite)

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PROBABILITY FUNCTIONS

• 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– P(E) = 1: the event is CERTAIN to occur– P(E) = 0: the event is certain NOT to occur

• Example: in the case of die casting, – 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

EXERCISES: ANALYTIC PROBABILITIES

• When we know the entire sample space, and we can assume that all outcomes are equally likely, we can compute the probability of events such as– P(1)– P(EVEN)– P(>3)

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PROBABILITIES AND RELATIVE FREQUENCIES

• 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

LOADED DICE

• The assumption that all outcomes have equal probability is very strong

• In most real situations (and with most real dice) probabilities of the outcomes are slightly different – P(1) = 1/4, P(2) = .15, P(3) = .15, P(4) = .15,

P(5) = .15, P(6) = .15

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JOINT PROBABILITIES• 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

• We use the same notation with WORDS: P(‘the’ & ‘car’)

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JOINT PROBABILITIES: TOSSING A DIE TWICE

• Sample space = { <1,1>, <1,2>, <1,3>, <1,4>, <1,5>, <1,6>, <2,1>, ….. ….. <6,1>, <6,2>, <6,3>, ……..}

EXERCISES: PROBABILITY OF TWO EVENTS

– P(first toss=1 & second toss=3)– P(first toss=even & second toss=even)

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OTHER COMBINATIONS OF EVENTS

• A B: either event A or event B happens– P(A B) = P(A) + P(B) – P(AB)– NB: If AB = , P(A ∅ B) = P(A) + P(B)

• A: event A does not happen– P( A) = 1 –P(A)

EXERCISES: ADDITION RULE

• P(A B) = P(A) + P(B) – P(AB)

• P( first toss = 1 second toss = 1)• P(sum of two tosses = 6 sum of two tosses =

3)

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PRIOR PROBABILITY VS CONDITIONAL PROBABILITY

• 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

ALL RACES

RACES WHEN IT RAINSRACES WON BY HARRY

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Conditional probability• Conditional probability is DEFINED as follows:

• 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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BP

BAPBAP

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EXAMPLE• Consider the case of Harry the horse again:

• Where:– P(WIN&RAIN) = 15/100 = .15– P(RAIN) = 30/100 = .30

• This gives:

• (in agreement with our intuitions)

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RAINP

RAINWINPRAINWINP

5.03.0

15.0)|( RAINWINP

EXERCISES

• P(sum of two dice = 3)• P(sum of two dice = 3 |

first die = 1)

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THE MULTIPLICATION RULE

• The definition of conditional probability can be rewritten as:– P(A&B) = P(A|B) P(B)– P(A&B) = P(B|A) P(A)

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INDEPENDENCE• 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)

EXERCISES

• P(H & H)• P(sum of two tosses greater than 6 &

first toss = 1)

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THE CHAIN RULE

• The multiplication rule generalizes to the so-called CHAIN RULE:– 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

NLE: – P(the big dog) = P(the) P(big|the) P(dog|the big)

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Bayes’ theorem• 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:

– P(YES|REL) = .95– P(YES| REL) = .005

• What is the probability that the document is relevant, when the system says YES?– P(REL|YES)?

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Bayes’ Theorem

• 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:– 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

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AP

BPBAPABP

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Application of Bayes’ theorem

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YESP

RELPRELYESPYESRELP

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RELPRELYESPRELPRELYESP

RELPRELYESP

002.099999.0005.000001.095.0

00001.095.0